This collection of texts explores the evolution of physics from classical mechanics to modern quantum theory. It begins by examining the limitations of classical physics when applied to phenomena at very high speeds or very small scales. The discussion covers topics like electromagnetism, relativity, and the concept of the aether, highlighting experimental evidence that challenged classical assumptions. It introduces special relativity with its postulates, consequences such as time dilation, and the role of the Lorentz transformation. Furthermore, the source addresses the development of quantum mechanics, including the wave-particle duality of light and matter, the Schrodinger equation, and its probabilistic nature. **Finally, it explores the structure of atoms, the quantization of energy, and the limits of absolute knowledge imposed by quantum mechanics.

Physics Study Guide: Vectors, Electromagnetism, Quantum Mechanics, and Thermodynamics

Quiz (Short Answer)

1. Explain the difference between a dot product and a cross product, including the type of result each produces. A dot product results in a scalar (a single number), calculated using the magnitudes of the vectors and the cosine of the angle between them. A cross product yields a vector, whose magnitude depends on the magnitudes of the vectors and the sine of the angle between them and whose direction is perpendicular to both original vectors.
2. State Newton’s First Law of Motion and briefly describe its significance. Newton’s First Law states that an object’s velocity remains constant unless acted upon by an external force. This law emphasizes the concept of inertia, establishing that objects naturally resist changes in their state of motion and requires an external influence to alter this state.
3. Summarize Gauss’s Law for electric fields and what it implies. Gauss’s Law for electric fields states that the divergence of the electric field is proportional to the charge density. This implies that electric fields originate from and terminate on electric charges, directly linking the presence of charge to the existence of an electric field.
4. What is the significance of Maxwell’s equations in understanding the nature of light? Maxwell’s equations demonstrate that light is an electromagnetic wave, a result derived from the equations in a region of space with no charges or currents. These equations link the behavior of electric and magnetic fields to the propagation of light, revealing its fundamental nature as an electromagnetic phenomenon.
5. Explain what is meant by the “lifetime” of an unstable particle. The lifetime of an unstable particle refers to the characteristic time after which approximately 63.2% of a sample of those particles will have decayed. It is also equivalent to the average time that any randomly selected unstable particle will exist before decaying.
6. Describe the key difference between relativistic and classical kinetic energy. Relativistic kinetic energy accounts for the increase in mass as an object approaches the speed of light, whereas classical kinetic energy assumes mass remains constant. At speeds much lower than the speed of light, the relativistic formula reduces to the classical formula (1/2)mv^2.
7. How does the ideal gas law connect microscopic properties to macroscopic properties? The ideal gas law (PV=nRT) relates the macroscopic properties of pressure, volume, and temperature to the microscopic properties of the number of moles of gas. Through classical physics, one can relate the average kinetic energy of gas molecules to temperature, thus connecting microscopic motion to macroscopic measurements.
8. Explain the concept of emissivity and its range of values. Emissivity describes a body’s efficiency in radiating or absorbing thermal radiation. Its value ranges from 0 (no emission) to 1 (perfect emission or a black body), indicating how effectively a surface emits energy compared to a theoretical perfect emitter.
9. What is the Balmer series, and what did it reveal about the hydrogen atom? The Balmer series is a set of visible spectral lines emitted by hydrogen, described by a specific mathematical relationship. It suggested that the energy levels within the hydrogen atom are quantized, leading to the emission of photons with only certain discrete wavelengths.
10. Explain how the de Broglie hypothesis contributed to the Bohr model of the atom. The de Broglie hypothesis, postulating that particles have wave-like properties, helped explain the Bohr model’s quantization of electron orbits. By suggesting that electron wavelengths must be integer multiples of the orbital circumference, it provided a physical basis for the allowed energy levels in the atom.

Quiz Answer Key

1. A dot product results in a scalar (a single number), calculated using the magnitudes of the vectors and the cosine of the angle between them. A cross product yields a vector, whose magnitude depends on the magnitudes of the vectors and the sine of the angle between them and whose direction is perpendicular to both original vectors.
2. Newton’s First Law states that an object’s velocity remains constant unless acted upon by an external force. This law emphasizes the concept of inertia, establishing that objects naturally resist changes in their state of motion and requires an external influence to alter this state.
3. Gauss’s Law for electric fields states that the divergence of the electric field is proportional to the charge density. This implies that electric fields originate from and terminate on electric charges, directly linking the presence of charge to the existence of an electric field.
4. Maxwell’s equations demonstrate that light is an electromagnetic wave, a result derived from the equations in a region of space with no charges or currents. These equations link the behavior of electric and magnetic fields to the propagation of light, revealing its fundamental nature as an electromagnetic phenomenon.
5. The lifetime of an unstable particle refers to the characteristic time after which approximately 63.2% of a sample of those particles will have decayed. It is also equivalent to the average time that any randomly selected unstable particle will exist before decaying.
6. Relativistic kinetic energy accounts for the increase in mass as an object approaches the speed of light, whereas classical kinetic energy assumes mass remains constant. At speeds much lower than the speed of light, the relativistic formula reduces to the classical formula (1/2)mv^2.
7. The ideal gas law (PV=nRT) relates the macroscopic properties of pressure, volume, and temperature to the microscopic properties of the number of moles of gas. Through classical physics, one can relate the average kinetic energy of gas molecules to temperature, thus connecting microscopic motion to macroscopic measurements.
8. Emissivity describes a body’s efficiency in radiating or absorbing thermal radiation. Its value ranges from 0 (no emission) to 1 (perfect emission or a black body), indicating how effectively a surface emits energy compared to a theoretical perfect emitter.
9. The Balmer series is a set of visible spectral lines emitted by hydrogen, described by a specific mathematical relationship. It suggested that the energy levels within the hydrogen atom are quantized, leading to the emission of photons with only certain discrete wavelengths.
10. The de Broglie hypothesis, postulating that particles have wave-like properties, helped explain the Bohr model’s quantization of electron orbits. By suggesting that electron wavelengths must be integer multiples of the orbital circumference, it provided a physical basis for the allowed energy levels in the atom.

Essay Questions

1. Discuss the limitations of classical physics in explaining phenomena at the atomic level and how quantum mechanics addresses these limitations. Provide specific examples from the source material.
2. Explain how Maxwell’s equations revolutionized our understanding of electromagnetism and the nature of light. What are the key implications of these equations?
3. Describe the Bohr model of the atom and its significance in the development of quantum theory. How did the de Broglie hypothesis contribute to a more complete understanding of the atom?
4. Explain the concept of wave-particle duality and provide experimental evidence that supports this concept.
5. Discuss the relationship between temperature and the average kinetic energy of the constituents of a material body, and how this relationship leads to an understanding of heat energy transfer.

Glossary of Key Terms

• Vector: A quantity with both magnitude and direction.
• Magnitude (of a vector): The length or size of a vector.
• Unit Vector: A vector with a magnitude of one, used to indicate direction.
• Dot Product: A scalar quantity resulting from the multiplication of two vectors, calculated as the product of their magnitudes and the cosine of the angle between them.
• Cross Product: A vector quantity resulting from the multiplication of two vectors, with a magnitude equal to the product of their magnitudes and the sine of the angle between them, and a direction perpendicular to both vectors.
• Newton’s First Law: An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
• Charge Density: The amount of electric charge per unit volume.
• Electric Field: A region around an electric charge where a force would be exerted on other charges.
• Magnetic Field: A region around a moving electric charge where a force would be exerted on other moving charges.
• Maxwell’s Equations: A set of four fundamental equations describing the behavior of electric and magnetic fields, and how they are related to electric charges and currents.
• Gauss’s Law (Electric): The total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity.
• Gauss’s Law (Magnetic): Magnetic monopoles do not exist; the total magnetic flux out of a closed surface is zero.
• Faraday-Maxwell Law: A time-varying magnetic field creates an electric field.
• Ampere-Maxwell Law: A magnetic field is created by electric current, or by a changing electric field.
• Lifetime (of a particle): The average time an unstable particle exists before decaying.
• Relativistic Kinetic Energy: The kinetic energy of an object, taking into account relativistic effects at high speeds, where mass increases with velocity.
• Ideal Gas Law: An equation of state relating pressure, volume, temperature, and the number of moles of an ideal gas: PV = nRT.
• Emissivity: The ratio of energy radiated by a material to energy radiated by a black body at the same temperature.
• Black Body: An idealized object that absorbs all incident electromagnetic radiation and emits radiation based only on its temperature.
• Spectral Radiance: The power emitted per unit area per unit solid angle per unit wavelength.
• Balmer Series: A set of spectral lines in the visible region of the electromagnetic spectrum emitted by hydrogen atoms.
• De Broglie Hypothesis: The concept that all matter has wave-like properties, with a wavelength inversely proportional to its momentum.
• Bohr Model: A model of the atom in which electrons orbit the nucleus in specific, quantized energy levels.
• Quantization: The concept that energy, momentum, and other physical quantities can only exist in discrete values.
• Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying i2 = −1.
• Complex Conjugate: For a complex number a + bi, its complex conjugate is a – bi.
• Schrödinger Equation: A fundamental equation in quantum mechanics that describes the time evolution of a quantum mechanical system.
• Postulates of Quantum Mechanics: The fundamental assumptions that underlie quantum mechanics, including the representation of the state of a system by a wave function, the representation of observable quantities by operators, and the probabilistic nature of measurement outcomes.
• Operator (quantum mechanics): A mathematical function that when applied to a quantum state (wavefunction) yields another state and can represent a physical observable.
• Eigenvalue: A characteristic number associated with an operator; the only possible result of a measurement of an observable quantity in quantum mechanics.

Physics: From Mechanics to Quantum Theory

Okay, here’s a briefing document summarizing the key themes and ideas from the provided excerpts.

Briefing Document: Key Concepts in Mechanics, Electromagnetism, and Quantum Mechanics

Overview:

The provided excerpts cover fundamental concepts in physics, spanning classical mechanics, electromagnetism, and the introduction of quantum mechanical ideas. The text progresses from basic vector operations and Newtonian mechanics to Maxwell’s equations and then to the breakdown of classical physics in the realm of unstable particles, radiation, and atomic structure. It then introduces core principles of quantum mechanics and some applications. A recurring theme is the tension between classical and quantum descriptions of the universe and how the limitations of classical physics necessitate the development of quantum theory.

Key Themes and Ideas:

1. Vector Operations and Newtonian Mechanics:

• Vectors: Vectors are defined with magnitude and direction, broken down into components. Unit vectors (I, J, K) denote directions along coordinate axes. “Vectors even though they carry both length information and direction information can be summarized as having a singular length that characterizes the full straight line distance that you would have to go to get from the beginning of the vector to the end of the vector and this is known as its length or its magnitude.”
• Vector Addition/Subtraction: Vector components are added or subtracted to find the resultant vector.
• Dot Product: Produces a scalar. C = A · B = |A||B|cos(θ). “In component notation you can calculate this by taking the X components and multiplying them together together taking the Y components and multiplying them together Etc and then adding all of those products together and again this yields a pure scalar a pure number with no direction.”
• Cross Product: Produces a vector, perpendicular to both input vectors. A x B = C, where |C| = |A||B|sin(θ).
• Newton’s Laws: The first law states that objects maintain constant velocity unless acted upon by an external force. “The first law states that the state of motion that is the velocity of an object remains constant unless the object is acted upon by an external Force absent external influences the natural state of an object is to maintain whatever velocity it presently has.” ΣF = 0 implies a = 0.

1. Electromagnetism and Maxwell’s Equations:

• Electric and Magnetic Fields: Electric charge density is the source of electric fields, while electric current density (moving charges) is the source of magnetic fields. “A density of electric charge however is the source of the electric field of force Mass has nothing to do with the electric field of force… now an electric current density that is a flow of electric charge is the source of a magnetic field of force.”
• Nabla Operator (∇): A vector operator made of derivatives with respect to space, used to express Maxwell’s equations concisely. “I’m going to define a symbol it’s this funny triangular symbol known as nabla because it resembles a ancient harp of the same name…it’s made of derivatives.”
• Maxwell’s Equations: Four equations governing electric and magnetic fields.
• Gauss’s Law for Electric Fields: Relates charge density to the electric field. “a charge density that is a charge per unit volume row is the source of an electric field on the left hand side we have this operator I defined above which is just a triplet of space derivatives acting on an electric field via the action of the dot product so this thing returns a number and that number is equal to the charge density divided by Epsilon knot which is a constant of nature”
• Gauss’s Law for Magnetic Fields: Indicates no magnetic monopoles. “What this equation tells you is that so far as we know there are no such thing as a magnetic charge”
• Faraday-Maxwell Law: A time-changing magnetic field generates an electric field. “The Faraday Maxwell law tells me that if I have a time-changing magnetic field this can generate an electric field”
• Ampere-Maxwell Law: A time-changing electric field or current density generates a magnetic field.
• Light as Electromagnetic Waves: Maxwell’s equations explain light as an electromagnetic wave. By solving Maxwell’s equations in empty space (no charges, no currents), wave-like solutions for electric and magnetic fields are obtained.

1. Relativistic Kinematics

• Relativistic Momentum and Energy: The text addresses relativistic corrections to kinetic energy and momentum, “these quantities for momentum and kinetic energy have all the right behaviors they don’t look like what they looked like in their assumed classical forms they reduce to their classical forms in the appropriate limit and they leave laws of physics invariant where they can be applied”
• Kinetic Energy: An equation for relativistic kinetic energy is presented: Kinetic energy of a particle is simply given by the quantity of its gamma Factor minus 1 times mc^2.”

1. Unstable Particles and Decay:

• Radioactive Decay: Mathematics of unstable particles was developed to understand radioactive decay.
• Half-life and Lifetime: The concept of half-life (T1/2) and characteristic lifetime (τ) is introduced to describe the exponential decay of unstable particles. “If you allow enough time to pass that one time constant’s worth of time goes by you find that 63.2% of the original number of objects are gone.” τ is the time constant and also the average lifetime of an unstable particle. “The halflife is equal to the time constant time the natural log of 2”
• Exponential Decay: The number of unstable objects decreases exponentially with time: N(t) = N0 * e^(-λt), where λ = 1/τ.

1. Heat, Radiation, and Blackbody Radiation:

• Heat Transfer: Heat can be transferred through conduction, convection, and radiation.
• Stefan-Boltzmann Law: The power radiated/absorbed by a body is proportional to its emissivity, surface area, and the fourth power of its temperature: P = εσAT^4. “The power radiated or absorbed by a body that that is to say the change in heat energy per unit change in time is given by the product of four numbers Sigma which is a constant of nature known as the Stefan boltzman constant… the stuff on boltzman constant is multiplied by another number which is this curly lowercase Greek Epsilon Epsilon is the emissivity of the surface of a body”
• Blackbody Radiation: A perfect emitter and absorber of radiation (ε = 1).
• Spectral Radiance: The power emitted per unit wavelength, denoted as B(λ).
• Rayleigh-Jeans Law: A classical attempt to predict spectral radiance, which fails at short wavelengths (the “ultraviolet catastrophe”). “for say a spherical body that’s heated to uh a certain temperature T and that body has a certain surface area a the shorter the wavelength of the radiation you consider being emitted from the body the more and more power is radiated around that wavelength if true this would be a catastrophic feature of nature”.
• Planck’s Law: Provides the correct description of blackbody radiation, demonstrating the quantization of energy. Plank’s law Nails it Max Plank’s law as he derived it in the early 1900s was the Cornerstone of the correct description of the radiation from heated matter”

1. Ideal Gas Law and Kinetic Theory:

• Ideal Gas Law: PV = nRT, relates pressure, volume, number of moles, ideal gas constant, and temperature.
• Molar Mass: Molar mass is defined as the mass per mole (mass per 6.022×10^23 molecules/atoms).
• Kinetic Theory: Connects microscopic properties (molecular speed, mass) to macroscopic properties (pressure, temperature). The text details how force and pressure can be related to the average kinetic energy of gas molecules.
• Average Kinetic Energy: Kavg = (3/2)kT, where k is the Boltzmann constant (R/Na). “… the average kinetic energy of a constituent of an ideal gas is given simply by a number three halves times another number the gas constant divided by avagadro’s number times a single variable the temperature of that gas”

1. Atomic Structure and Quantum Mechanics:

• Atomic Emission Spectra: Excited atoms emit discrete lines of light, unique to each element. These lines could not be explained by classical physics. “Atoms in general when excited by an ionizing electric potential emit not a continuous rainbow of colors but rather a discrete set of color colors”
• Balmer Series: A mathematical relationship between the wavelengths of visible light emitted by hydrogen. Balmer noted that the wavelength of each of these lines is given by a simple formula a constant btimes this ratio an integer n^ 2 / by that same integer n^ 2us 2^ 2 or 4″
• Early Atomic Models:Thomson Model: Electrons embedded in a uniform positive charge (“plum pudding”).
• Rutherford Model: A central, positively charged nucleus surrounded by orbiting electrons. This was based on alpha particle scattering experiments.
• Bohr Model:Quantization of Angular Momentum: Bohr postulated that angular momentum is quantized (L = nħ). “Since h and H bar the reduced plunks constant have units of angular momentum that is jewles time seconds it might be in an atom that the angular momentum L is a multiple an integer multiple of H bar”
• Matter Wave Hypothesis: de Broglie’s hypothesis (λ = h/p) implies that only certain wavelengths are allowed for an electron orbiting the nucleus.
• Allowed Radii and Energies: Combining the quantization condition with classical mechanics leads to discrete allowed radii (rn = n^2a0) and energy levels (En = -13.6 eV / n^2). ” the fact that just using a classical model of the atom combined with matter wave nature of the electron one can immediately reproduce a pattern in the world around you in this case the Balmer series of atomic emission spectrum lines this is incredible”

1. Simple Harmonic Motion and Waves

• SHM: a model of oscillatory behavior.
• Wave Equation: A mathematical description of wave behavior relating spatial and temporal derivatives of the wave amplitude.

1. Complex Numbers and their Use in Describing Waves:

• Complex Numbers: The text introduces complex numbers (Z = x + iy) and the imaginary unit (i = √-1). It explains how to find the magnitude of a complex number using the complex conjugate (Z* = x – iy): |Z|^2 = Z*Z = x^2 + y^2. “To get a real number you need to do something like this and this is part of what defines the algebra of complex numbers you’re going to take Z and you’re going to multiply it by a special version of itself known as Zar uh this is just x + i y the original complex number times x minus i y”
• Euler’s Formula: The excerpts allude to but don’t explicitly state Euler’s formula (e^(ix) = cos(x) + i sin(x)), which is used to simplify wave equations. It also sets up the use of complex conjugates with respect to calculations of wavefunctions.

1. Quantum Mechanics Postulates

• The state of a system is completely described by a wavefunction.
• Every observable property of the system has a corresponding mathematical “operator”. (e.g. energy is measured by a Time derivative).

Quotes Illustrating Key Shifts in Understanding:

• “…but the real laws of nature that we encounter in a course on introductory mechanics are Newton’s famous three laws of motion…” – This highlights the starting point of classical mechanics.
• “Now what’s amazing about the laws of electricity and magnetism Maxwell’s equations is that when you consider them in a particular situation it finally clarifies what the heck the nature of light is…” – This marks a transition to electromagnetism and its explanatory power.
• “If true this would be a catastrophic feature of nature…” – This reflects the crisis in classical physics when applied to blackbody radiation.
• “They are all solutions to a wave equation that is an equation that describes how changes in space relate to changes in time.”

Conclusion:

The excerpts provide a concise overview of fundamental physics concepts, from basic mechanics to the dawn of quantum theory. The progression highlights how classical physics, while successful in many domains, breaks down when applied to phenomena at very small scales, requiring the development of quantum mechanics. The discussion of the blackbody problem and atomic structure exemplifies this shift in paradigm. The introcuction of complex numbers provides a basic tool for QM, and the postulates begin to lay out the mathematical structure for the theory.

Vectors, Fields, and Particle Physics: Key Concepts

Vectors and Unit Vectors

Q1: What are vectors and how are they represented? Vectors are quantities that have both magnitude (length) and direction. They are typically represented with an arrow over the variable, such as ‘a’. In a Cartesian coordinate system, a vector can be broken down into components along the x, y (and z in 3D) axes (Ax, Ay, Az). Unit vectors (like i-hat, j-hat, and k-hat) denote direction along the x, y, and z axes, respectively.

Vector Magnitude and Operations

Q2: How is the magnitude (or length) of a vector calculated? The magnitude of a vector ‘a’ can be denoted as ‘a’ (without the arrow) or as |a|. It is calculated using the Pythagorean theorem in multiple dimensions. The square root of the sum of the squares of its components (e.g., √(Ax^2 + Ay^2)). Q3: How are vectors added and subtracted? Vectors are added component-wise. If you have vectors ‘a’ and ‘b’, the resulting vector ‘c’ (c = a + b) has components Cx = Ax + Bx, Cy = Ay + By, and Cz = Az + Bz. Subtraction follows the same principle, just with subtraction instead of addition. Q4: What are dot product and cross product and what do they return? The dot product (a · b) is a scalar (a single number) quantity calculated as the magnitude of a times the magnitude of b times the cosine of the angle between them. In component form, it’s (Ax * Bx) + (Ay * By) + (Az * Bz). The cross product (a × b) yields another vector ‘c’, perpendicular to both ‘a’ and ‘b’. The magnitude of the result is the magnitude of a times the magnitude of b times the sine of the angle between the two vectors.

Force Fields and Maxwell’s Equations

Q5: How are electric and magnetic fields created, and what are Maxwell’s equations? Electric fields are created by electric charge density (charge per unit volume), while magnetic fields are created by a flow of electric charge, aka current density. Maxwell’s equations are a set of four equations that describe these fields. They relate the electric and magnetic fields to their sources (charges and currents) and to each other:

1. Gauss’s law for electric fields
2. Gauss’s law for magnetic fields
3. Faraday-Maxwell law
4. Ampere-Maxwell law These equations also clarify the nature of light as electromagnetic waves.

Unstable Particles and Half-Life

Q6: What is the relationship between lifetime, time constant, and half-life for an unstable particle? Unstable particles decay over time, described by exponential decay. The lifetime (τ) is the average time an unstable particle exists. The time constant (τ) is the time it takes for approximately 63.2% of a sample of unstable particles to decay. The half-life (t1/2) is the time it takes for half of the particles to decay. The relationship is t1/2 = τ * ln(2).

Relativistic Kinetic Energy

Q7: How is relativistic kinetic energy calculated, and how does it relate to classical kinetic energy? Relativistic kinetic energy (K) is given by K = (γ – 1)mc^2, where γ is the Lorentz factor (gamma), m is mass, and c is the speed of light. In the limit where velocity (u) is much smaller than the speed of light, this equation reduces to the classical kinetic energy formula K = (1/2)mu^2.

Ideal Gas Law and Kinetic Theory

Q8: How does the ideal gas law relate macroscopic properties to microscopic properties of gas molecules? The ideal gas law (PV = nRT) relates the pressure (P), volume (V), number of moles (n), ideal gas constant (R), and temperature (T) of a gas. From that law it can be derived that the average kinetic energy of a single gas molecule is directly proportional to the temperature of the gas (KE = (3/2)kT), where k is the Boltzmann constant. Furthermore, the average speed of the gas molecules relates to the temperature and molar mass of the gas.

Understanding the Speed of Light: Definition, Constancy, and Implications

The speed of light is a crucial concept in modern physics, particularly in the context of special relativity. Here’s a breakdown:

• Definition: The speed of light refers to the rate at which light travels a certain distance in a specific amount of time. It is the number of meters that light can travel once emitted by a source in a certain amount of time.
• Numerical Value: Light travels at 299,792,458 meters per second. This is often approximated as 2.998 * 10^8 m/s.
• Constancy: The speed of light is constant for all observers, regardless of their state of motion or the motion of the light source. This principle is a cornerstone of Einstein’s theory of special relativity and was established through experiments like the Michelson-Morley experiment.
• Implications for Space and Time: The constant speed of light leads to some surprising consequences:
• Relativity of time and space: Measurements of space and time are not absolute and can differ for observers in different states of motion.
• Time dilation: Observers in motion relative to others will experience a slower passage of time.
• Length contraction: Objects in motion will appear to be shorter in the direction of motion to a stationary observer.
• Notion of simultaneity: Events that appear simultaneous in one frame of reference may not be simultaneous in another.
• Historical context:
• Early attempts at measurement: Galileo attempted to measure the speed of light using lanterns and assistants on distant hills, but the speed of light was too fast for his methods.
• Rømer’s determination: Ole Rømer used observations of Jupiter’s moon Io to make a reasonable determination that light travels at a finite speed.
• Modern establishment: By the time of Einstein’s publications, the speed of light had been established to be within approximately 50 km/s of today’s methods.
• Practical Considerations: A useful rule of thumb is that light travels approximately one foot in one billionth of a second (one foot per nanosecond).

The speed of light, therefore, is not just a measure of how fast light travels, but a fundamental constant that shapes our understanding of space, time, and the universe.

Understanding Wave Phenomena: Properties, Effects, and Duality

Wave phenomena are characterized by oscillatory behavior distributed in space and time. Here’s an overview of key concepts related to wave phenomena, as discussed in the sources:

• Gross properties of waves: To understand waves, it’s helpful to focus on a specific point and observe its repetitions, which represent the wave’s spatial or temporal distribution.
• Wavelength ((\lambda)): The distance between two corresponding points on a wave, such as the distance between crests or the distance between maximum displacement from zero.
• Period (T): The time it takes for a wave to complete one full cycle, or the time between successive maximums or minimums passing a fixed point in space.
• Frequency (f or (\nu)): The number of cycles per unit of time, typically measured in Hertz (Hz) or per second, and is the inverse of the period ((f = 1/T)).
• Wave front: A line or plane that indicates the location in space of a maximum of the traveling wave.
• Speed of a wave: The speed ((v)) at which a wave moves through space is the product of its wavelength and frequency ((v = \lambda f)). For light waves in a vacuum, this speed is the speed of light ((c)), approximately (2.998 \times 10^8) m/s.
• Doppler effect: The change in frequency or wavelength of a wave for an observer who is moving relative to the wave source.
• Classical Doppler Effect: Occurs when a source emitting a wave is moving relative to an observer. For example, if a car is moving and honking its horn, the sound waves in front of the car will be compressed (shorter wavelength, higher frequency), while the sound waves behind the car will be stretched out (longer wavelength, lower frequency).
• Relativistic Doppler Effect: A modification of the classical Doppler effect that accounts for the effects of special relativity on space and time measurements.
• Wave equations: Mathematical expressions describing how changes in space relate to changes in time for a wave. Examples include:
• Mechanical waves: For a vibrating string, the vertical displacement ((Y)) varies with horizontal position ((x)) and time ((t)), described by a wave equation involving second derivatives with respect to both space and time.
• Electromagnetic waves: Maxwell’s equations lead to a wave equation describing the propagation of oscillating electric and magnetic fields in space.
• Energy in waves:
• Mechanical waves: The kinetic and potential energy of a small segment of a vibrating string can be calculated, with the total energy being constant and dependent on factors like angular frequency, linear mass density, and displacement.
• Electromagnetic waves: The energy of a light quantum (photon) is related to its frequency by (E = hf), where (h) is Planck’s constant. The momentum ((p)) of a photon is related to its wavelength by (p = h/\lambda).
• Wave-particle duality: Light and matter exhibit both wave-like and particle-like properties.
• Light: Demonstrates wave-like behavior through phenomena like diffraction and interference, but also exhibits particle-like behavior as seen in the photoelectric effect and black body radiation.
• Matter: Particles like electrons can also exhibit wave-like behavior, as demonstrated by electron diffraction experiments. De Broglie proposed that the wavelength of a matter wave is inversely proportional to its momentum: (\lambda = h/p).
• Interference: When waves overlap, they can either constructively interfere (amplitudes add up) or destructively interfere (waves cancel out). A classic example is the double-slit experiment, where light or matter waves passing through two slits create an interference pattern of bright and dark spots.
• Scale and wave behavior: The manifestation of wave or particle behavior depends on the size of the system relative to the wavelength of the radiation.
• If the wavelength is much larger than the system, wave-like behavior dominates.
• If the wavelength is much smaller than the system, particle-like behavior dominates.
• Matter waves: de Broglie’s hypothesis suggests that matter also has wave properties, with the energy and momentum of a particle related to the frequency and wavelength of its corresponding matter wave. This is described by (E = hf) and (p = h/\lambda).

Understanding Relative Motion: Classical and Special Relativity

Relative motion is a concept that describes how the motion of an object is perceived differently depending on the observer’s own motion. It’s a fundamental aspect of both classical and modern physics and is essential for understanding how different observers can disagree on the measurements of space and time.

Here’s an overview of relative motion, as discussed in the sources:

• Classical (Galilean) Relativity:
• Basic principle: In introductory physics, relative motion is often introduced with examples like a person on a moving train throwing a ball. Observers in different frames of reference (e.g., the person on the train and someone standing on the ground) will perceive the motion of the ball differently.
• Galilean transformation: This set of equations mathematically relates spatial coordinates and velocities of objects from a frame considered at rest to a frame considered moving, assuming time passes the same for all observers.
• Intuitive but limited: Classical physics and Galilean relativity provide an intuitive understanding of relative motion at everyday human scales and speeds.
• Ether: In classical physics, it was presumed that light, like mechanical waves, needed a medium to travel through, called the ether. The speed of light was thought to be constant with respect to this ether.
• Einstein’s Theory of Special Relativity:
• Postulates: Einstein’s theory is based on two key postulates:
• The laws of physics are the same for all observers, regardless of their state of relative motion.
• The speed of light in a vacuum is the same for all observers, regardless of their state of motion or the motion of the light source.
• Rejection of Absolute Time: Einstein challenged the idea that time is experienced the same way by all observers.
• Consequences:
• Time dilation: Moving observers experience time passing more slowly relative to stationary observers.
• Length contraction: Objects in motion appear shorter in the direction of motion to a stationary observer.
• Relativity of simultaneity: Events that are simultaneous in one frame of reference may not be simultaneous in another.
• Frames of Reference:
• A frame of reference is any object or system, all of whose parts move at the same velocity with respect to an agreed-upon reference point in space.
• Descriptions of events depend on the frame of reference in which the observation is made.
• It is impossible to determine whether you are in motion by making observations in your frame of reference.
• Velocity Addition:
• Non-intuitive at high speeds: In special relativity, velocities do not simply add linearly as they do in classical mechanics.
• Velocity Transformation Equation: Provides a way to correctly add velocities, ensuring that the speed of light remains constant for all observers.
• Light Speed Limit: Nothing can move faster than the speed of light.
• Practical Implications and Examples:
• Muon Decay: The behavior of muons, subatomic particles with short lifetimes, provides experimental evidence for time dilation and length contraction. Muons created in the upper atmosphere travel much farther than expected because their lifetimes are extended due to their high speeds.
• Doppler Effect: The change in frequency or wavelength of a wave (e.g., light) due to the relative motion between the source and the observer. This effect is used to infer relative velocities on a cosmic scale.

Understanding relative motion requires careful consideration of the observer’s frame of reference and the effects of special relativity, especially when dealing with high speeds. The speed of light remains the same for all observers. The laws of physics remain the same for all observers.

General Relativity: Spacetime, Gravity, and Einstein’s Theory

General relativity is a theory of space, time, and gravitation developed by Albert Einstein that extends his earlier theory of special relativity. It provides a framework for understanding gravity not as a force, but as a consequence of the curvature of spacetime caused by mass and energy.

Here’s an overview of general relativity, based on the information in the sources:

• Transition from Special Relativity:
• Einstein sought to generalize special relativity into a complete theory of space and time, aiming to explain gravity.
• This endeavor required a decade of struggle and relearning advanced mathematics.
• Key Concepts:
• Spacetime: General relativity treats space and time as a single, four-dimensional framework called spacetime.
• Curvature of Spacetime: Mass and energy cause spacetime to curve, bend, or warp.
• Gravity as Curvature: What we perceive as gravity is the result of objects following the curvature of spacetime. Objects are not pulled down by the mass of the Earth. Rather, they are following a path in SpaceTime that’s curved due to the presence of the mass energy of the earth bending that space and time.
• Equivalence Principle: There is no difference between an accelerated frame of reference and a frame experiencing a gravitational field.
• Implications and Predictions:
• Bending of Light: Gravity bends the path of light. The amount of bending predicted by general relativity is twice as large as that predicted by Newtonian physics.
• Gravitational Lensing: Massive objects can bend light from distant objects, allowing us to see objects that would otherwise be obscured. The degree of light deflection can be used to measure the mass of the intervening object.
• Gravitational Time Dilation: Time is affected by gravity. Clocks at different heights in a gravitational field run at different rates. This must be accounted for in systems such as the Global Positioning System (GPS).
• Cosmological Implications: General relativity is essential for understanding the origin and fate of the universe, including the Big Bang and the expansion of the universe.
• Inertial Mass vs. Gravitational Mass:
• Equivalence: General relativity is built on the observation that inertial mass (resistance to acceleration) and gravitational mass (response to gravity) are equivalent.
• Eötvös Experiment: Confirmed the equivalence of inertial and gravitational mass with great precision.
• Spacetime and Gravity:
• Space and time tell energy and matter how to move.
• Energy and matter tell space and time how to bend or curve.
• Mathematical Complexity:
• General relativity relies on complex mathematics beyond the scope of introductory physics courses.

Matter Waves: Wave-Particle Duality and Quantum Mechanics

Matter waves describe the wave-like behavior of particles, a concept that revolutionized physics in the early 20th century. This idea, primarily associated with Louis de Broglie, suggests that matter exhibits both particle-like and wave-like properties.

Here’s a detailed discussion of matter waves, based on information from the sources:

• De Broglie’s Hypothesis:
• In his 1924 PhD thesis, Louis de Broglie postulated that matter, like light, has wave properties.
• He proposed that the energy ((E)) and momentum ((p)) of a particle are related to the frequency ((f)) and wavelength ((\lambda)) of its corresponding matter wave through the same equations that apply to photons:
• (E = hf)
• (p = h/\lambda)
• Where (h) is Planck’s constant.
• Experimental Verification:
• De Broglie’s hypothesis was experimentally verified through scattering experiments, where matter, such as electrons, is scattered off a target.
• These experiments revealed interference patterns, demonstrating the wave nature of matter.
• For instance, when electrons are scattered off crystals, the resulting diffraction patterns are similar to those observed when light passes through a diffraction grating.
• These diffraction patterns could not be explained if electrons were solely particles.
• Wave-Particle Duality:
• Matter waves imply that particles, which had been primarily experienced as definite objects with specific locations in space and time, also exhibit wave-like behavior.
• The manifestation of wave or particle behavior depends on the scale of the system and the wavelength of the matter wave.
• When the wavelength is much smaller than the size of the object it interacts with, the particle-like aspects of the matter are observed.
• When the wavelength is comparable to or larger than the size of the object, the wave-like aspects are observed.
• Implications for Atomic Structure:
• The concept of matter waves helps explain the discrete atomic spectra of elements.
• In the Bohr model of the atom, electrons orbiting the nucleus can only exist in specific energy levels.
• This quantization arises from the requirement that the electron’s matter wave must form a standing wave around the circumference of the orbit.
• Only certain wavelengths (and therefore energies) satisfy this condition, leading to the observed discrete emission spectra.
• Heisenberg Uncertainty Principle:
• The wave nature of matter imposes a limit on the precision with which certain pairs of physical properties, such as position and momentum, can be known.
• This is formalized in the Heisenberg Uncertainty Principle, which states that the more accurately the position of a particle is known, the less accurately its momentum can be known, and vice versa.
• This principle arises because determining position requires localizing the wave, while determining momentum requires knowing its wavelength (and thus its spatial distribution).
• Mathematical Description:
• Schrödinger Equation: The wave equation for matter is the Schrödinger equation. It describes how matter waves propagate and evolve over time.
• The solutions to the Schrödinger equation are wave functions, which are generally complex functions.
• The square of the magnitude of the wave function represents the probability density of finding the particle at a particular location in space and time.
• Born interpretation (not stated explicitly in the source but implied), which is the most practical interpretation, gives a probabilistic meaning to the square of the wave function.
• Analogy for Understanding:
• Consider the example of a beaker in a tank of water. When struck by a long wavelength, the beaker exhibits gradual motion. When struck by a short wavelength, the beaker is violently hit.
• This illustrates how a wavelength short compared to the target will exhibit particle-like behavior, whereas a wavelength long compared to the target will exhibit wave-like behavior.

The Original Text

[Music] hello I’m Professor Steven sakula and I’d like to welcome you to Modern physics at SMU modern physics is the first course past the two semester introductory sequence where we begin to get closer to the physical principles that are at play in the modern technological and scientific world now like all good Sciences physics Builds on the past discoveries that have driven the field forward to continue to move through the frontier of human knowledge and modern physics is going to challenge you to Step Beyond the comfortable confines of introductory physics and into the modern view of the universe particularly space time energy and matter which are the four subjects at the heart of the science known as physics now introductory physics tends to leave off at the end of electricity and magnetism and that period coincides with the end of the 19th century the late 1800s which was a feverish time of experimentation the foundations of space and time had been laid in the centuries before by people like Isaac Newton and the laws of electricity and magnetism which were relatively the New Kids on the Block as regards scientific law had been established firmly in the mid 1800s it was believed at the time that for the most part everything that needed to be known about the universe had been established and all that was really left was to sort out some lingering puzzles that hadn’t quite yet been fit into the framework of Newton’s mechanics and the laws of electricity and magnetism now one of those phenomena is light it’s fairly straightforward to make a light source in the modern world all we have to do for instance is take a sealed tube filled with gas for instance and expose it to a strong electric field a large electric potential and we can coax it into emitting light now light is of course all around us it’s what’s literally Illuminating the scene here but what light was was only thinly established by the end of the 1800s we’ll look at the foundations of light and electricity and magnetism and mechanics in this video but one of the puzzles that was left over at the end of the 19th century was why certain elements emitted certain kinds of light but not others so for instance this sealed tube here is containing hydrogen gas and you’ll notice that it gives off a fairly strong reddish color on the other hand if I replace it with another tube of gas this one of mercury vapor Mercury being a a metal the only Elemental metal that’s liquid at room temperature in this case sealed as a vapor in the tube you’ll notice that in this case we get a very strong blue light from this particular element if you study the Fingerprints of these light emissions very closely you’ll observe that they have strong colors in some places but not in others in what is known as their atomic spectrum why was that the mystery of atomic Spectra would only be fully understood in the early part of the 20th century with the Advent of what we now call Modern physics specifically quantum physics now another interesting phenomenon that had been observed in the 1800s but which was not fully understood had to do with electric currents so what I’m going to do here is using the tribo electric effect using friction to build up an electric charge on a piece of plastic and then placing that plas plastic in contact with a conductor so that it can soak up that excess charge you’ll see that I’ve now caused a net electric charge to sit on this myar material attached to this aluminum soda can well nothing dramatic here so far the charges can freely move on the conductor and in this case they don’t like to be near each other because they’re electrons and they all have the same electric charge and so they Rush as far apart as they can get while remaining on the conductor so they’re trapped on this conductor they can’t escape but they’ve done everything in their power to get away from each other and in the process they’ve exerted an electric force that mechanically causes the myar to spread out in space now while I’ve been talking the light in this room which comes from something like a dozen light fixtures in the ceiling has been blasting this metal and yet while there’s a breeze in the room that moves the myar sheets around nothing’s really draining the charge off of this we don’t see these sheets appreciably falling down so so we have many watts of natural light coming in from light sources here nothing happens but it was observed in the 1800s that if you expose certain metals to certain colors of light or even colors of light that are Beyond human perception like ultraviolet they’ll begin to allow an electric current to flow this is the so-called photoelectric effect and it was unexplainable by Newton’s mechanics and the laws of electricity and magnetism as they were established in the 1800s so all I have to do according to the photoelectric effect idea is take something that emits ultraviolet radiation in this case a sanitizing wand for a sink or a counter or a toilet or something like that just go ahead and switch it on and it will begin to emit ultraviolet C radiation or light down toward the surface of the table and if I move it over the aluminum can it will begin to almost instantaneous ously drain the electric charge off the myar and if I sit here long enough it will eventually pull almost all of that electric charge off the myar leaving it hanging back down in a more vertical position this is four watts of UV light compared with tens of watts of natural light coming from the light fixtures in the ceiling why why does the color of the light matter to this effect more than the intensity of the light that was a Myst left over at the end of the 1800s now another mystery which would ultimately combine to lead to a firmer understanding of matter energy space and time had to do with heat energy now heat energy is something that we will explore in this class if I light this burner on fire so that it emits a nice blue flame I can take a biim metallic Strip This is a strip of two metals bonded together back to back and and I can place it in the flame now we’ll we’ll take a look at some of mechanically what’s going on here later in the course when we establish the foundations of heat energy uh in about a month or so but if I leave this metal in the flame not only because does it begin to bend but after enough time it will also begin to glow on its own originally it had a silvery color room temperature but as I continue to expose it to the source of heat energy this Open Flame and heat energy is transferred into the metal eventually the metal begins to Glow of its own accord with its own light now this wasn’t such a mystery to physicists and chemists of the 1800s but what was a mystery had to do with the degree of absorption of energy and the degree of reemission of energy at other at other frequencies and wavelengths other kinds of light that you can see with your eye and the exact relationship between heat energy and temperature and the kinds of radiation that should be emitted from a heated body proved a real challenge to mechanics electricity and magnetism and the laws of heat energy transfer or thermodynamics that were also established in the 1800s so modern physics is your gateway into a world that’s more consistent with the kind of world we live in today not the world of the 17 and 1800s but the world of the 1900s the 20th century and now the 21st century the foundations that we will establish in this course will lay the groundwork for a variety of important technological revolutions non-invasive Imaging of the human body the harnessing of the energy at the heart of every atom the construction of semiconductor devices which revolutionized our ability to do computations quickly and efficiently and a host of other advancements whose root Roots were all laid down in a revolutionary period in the transition between the 1800s and the 1900s that led into the era of modern physics welcome to this course for the rest of the video we’ll do a foundations review of introductory physics to refresh your memory about the most Salient things from the past two semesters of material and then we’ll move on to the foundations of modern physics in this lecture we’ll re-explore the foundations of introductory physics the basic concepts that should have been communicated to you in the first two semesters of introductory physics physics Builds on the past like all Sciences the discoveries of an earlier era influence our understanding of new discoveries and how to adapt our mathematical descriptions of nature in order to describe what we know know from the past but include new observations that don’t quite fit into the original framework that we had developed the big picture that I want you to take away from this foundation’s lecture can be broken into four large Parts first of all a foundation of the physics that you have learned so far is Newton’s mechanics these are laws of motion they link forces that act on objects to changes in the states of motion of the those objects and a state of motion is characterized by the velocity of an object the laws of mechanics were first established by Isaac Newton in his foundational publication philosophia naturalis prinkipia Mathematica or the prinkipia published in 1687 this set of laws illuminates how velocity the state of motion of an object can be influenced by external forces and codifies mathematically using geometry algebra and the newly invented calculus the way in which you can describe the interaction of these things in order to understand the natural world what would also be developed over the following centuries were a series of what we now call conservation laws these are principles that establish that certain quantities appear to be conserved that is left unchanged even by complex phenomena in nature these include things like the total energy of a system including internal forms of energy like chemical energy the total linear momentum of a system and the total angular momentum of a system and for closed and isolated systems where no external forces especially of the non-conservative variety those that can’t store in release energy and some kind of potential in those systems conservation laws will absolutely hold and they were established through careful chemical and physical work up through the 1700s and they continued to be built on in work on heat energy in the 1800s heat energy and the laws that govern its transfer from the mechanical form to the thermal form will be Revisited in later lectures the second part of the foundations of modern physics the third key idea is Newton’s law of gravitation that is the law that relates the distance between material bodies and the force between them a force that requires no actual physical contact no medium to be present between two things in order for them to exert a force on each other and this was also established in Newton’s prinkipia and finally the last set of laws of physics that we have to accept as a foundation for what’s going to happen in this course are the laws of electromagnetism these are the rules of electricity and magnetism describing them as forces in the same way that gravity is a force that can induce changes in states of motion again without physical contact between material bodies electric forces and magnetic forces can operate even if there’s no medium between the two bodies that are interacting with each other via these forces they were established in the 17 to 1800s and they were finally codified formally in four equations known as Maxwell’s equations in 1862 one of the mathematical foundations of describing nature in physics is a kind of number known as a vector the these are essential to describing any multi-dimensional quantity and they have a well-defined algebra which you should have exercised in previous physics courses you probably have also exercised these in a dedicated math or engineering or both course vectors are numbers that can be built from scalers scalers are numbers that have no directional information so for instance a good example of a scaler would be if you asked for directions to somebody’s house and they told you go 10 miles well that might eventually by going 10 miles get you to their house but without some crucial directional information how far east south north or west should I go to add up to those 10 miles you’re probably not going to make the journey successfully scalers however can be assembled using for instance component notation into a vector so here for instance is demonstrated a vector denoted a with a little arrow over its head and it’s broken into components it has a component that lies entirely along the x coordinate axis in a cartisian coordinate system with length a with a subscript X and similarly it has a component along the Y AIS in a cartisian coordinate system a with a subscript Y and these little vectors here I with a little triangular hat over at J with a little triangular hat over it we’ll come back to those in a moment but they’re essential in indicating a dedicated Direction either only along the x or y or Z axis vectors even though they carry both length information and direction information can be summarized as having a singular length that characterizes the full straight line distance that you would have to go to get from the beginning of the vector to the end of the vector and this is known as its length or its magnet magnitude this can be denoted in one of several ways either just drawing the vector with no Arrow over it so a in this case or putting absolute value signs around the vector that’s another common notation for length or magnitude of a vector and this can be computed using the sums of the squares of the components and then you take the square root of that total sum in two Dimensions this will recall the familiar Pythagorean theorem which given the lengths of of the sides of a right triangle will tell you the length of the hypotenuse then there are unit vectors this is a a subspecies of vector and they’re special because they are vectors whose length is always exactly one in whatever unit system you choose to use unit vectors are denoted with that little triangular hat symbol so for instance I hat J hat and K hat as they would be denoted in spoken terms are special and they’re unit vectors that point only along the X Y and Z axis respectively of a cartisian coordinate system this also means that because the angles between the X and Y Y and Z and z and x axis are 90° the angles between these unit vectors are also always 90° for any pair you can add vectors so for instance if I have a vector a and a vector B and I want to know what the resulting Vector for in C with a little Vector Arrow over its head looks like all I have to do is take the X components and add them together noting that they point along the IAT Direction take the Y components and add them together noting that they point along the y direction Etc and this will give you the resulting sum of two vectors you can replace the sum with a minus sign to get the difference of two vectors but the math is the same there are two kinds of multiplicative products of vectors the dot product which gives you a number and the cross product which returns a vector the dot product is given by the following notation C can be represented as the dotproduct of two vectors A and B with a little dot between them and it’s a number it’s a pure scalar whose size is the magnitude of a Time the magnitude of B time the cosine of the angle between a and b in component notation you can calculate this by taking the X components and multiplying them together together taking the Y components and multiplying them together Etc and then adding all of those products together and again this yields a pure scalar a pure number with no direction on the other hand the cross product the other multiplicative operation between two vectors yields a vector so in this case the cross product of two vectors A and B would yield a third Vector C the cross product is denoted by putting a cross multiplicative sign in between the two vectors A and B this one’s a little bit more complicated and you have to be a bit more careful with this I like component notation because you can essentially distribute the multiplication algebraically between the two vectors A and B and you wind up with terms that look like the X component of a and the Y component of b with this cross product of unit vectors next to it and then the Y component of a and the X component of b with the reverse cross product of I and J hat next to it and then a bunch of other terms that look similar to this depending on how many dimensions this thing has and in the end this yields a pure Vector with a length given by the magnitude of a Time the magnitude of B time the sign of the angle between the two vectors A and B also the vector c will always point at exactly a right angle to both a vector and B Vector that’s one of the natural consequences of the cross product now the cross products of coordinate axis unit vectors like IH hat J hat and K hat obey the following rules the cross product of any unit Vector with itself is zero because there is no Vector that’s perpendicular at the same time to both I and itself there’s an infinite number of those vectors and the cross product yields a result of zero for this similarly with J cross J and K cross K now the rule of thumb for computing all of the other cross products is that I cross J is K and then if you kind of conveyor belt K to the beginning of this operation move I to where J is and move j to where K is you get one of the other cross products K cross I is J and then similarly doing this conveyor belt permutation one more time socalled cyclic permutation you get J cross K is I now what about J cross I I cross K or k cross J well if you swap the order on the left side of these equations then the right side changes by a minus sign so J cross I would be negative Kat I cross K would be negative J hat and so forth vectors are an essential building block of everything that happens in mechanics but the real laws of nature that we encounter in a course on introductory mechanics are Newton’s famous three laws of motion the first law states that the state of motion that is the velocity of an object remains constant unless the object is acted upon by an external Force absent external influences the natural state of an object is to maintain whatever velocity it presently has this can be summarized in an equation as follows the sum of all forces with subscript I and there can be from 1 2 3 all the way up to capital N forces acting on an object if all of those add up and cancel each other out so that there is zero net force acting on an object then the resulting acceleration that is the change of velocity with respect to time or the change of the state of motion with respect to time given by the second derivative of a position Vector of the object is zero no net force no change in state of motion the more general form of this equation is given by Newton’s Second Law which relates the net unbalanced force acting on an object to any resulting acceleration or change in the state of motion of that object the change in Newton’s second law is proportional to something force and acceleration can be related to each other by a simple equation and the constant of proportionality between force F and acceleration a is given by m the so-called inertial mass of an object because you can write the acceleration as the second derivative with respect to time of the position Vector of an object I’ve put here the calculus notation for the acceleration in three dimensions where R Vector is a position Vector X Y and Z that not only can change with time but whose change with time can be further altered by having an external force act on it that is an acceleration and then finally there’s Newton’s third law that in every interaction of two material objects let’s call them A and B two forces are in action the direction of the force exerted by object a on object B is the opposite of the force of object B on object a but they are otherwise equal in magnitude so if I take my hand and push on the surface of a table the table pushes back against my hand with an equal magnitude but opposite direction Force that’s why my hand doesn’t go through the table now usually after learning about Newton’s Laws of Motion we then learn about quantities that are associated with motion these are known as energy and momentum what what is common between these quantities is that they vary in some proportion to the degree of motion so for example the quantity of energy associated with a moving object so-called kinetic energy is proportional to mass and to the square of the velocity of an object it is a scalar because you square the velocity you lose all directional information about it and the exact equation for kinetic energy is determined to be 1/2 * the mass times the velocity squared or the speed squared of an object there is a Direction full quantity of motion and that is known as linear momentum it is proportional to mass and directly to the velocity of a body at least in this classical physics and this introductory mechanics we learn about this is observed to be the thing that appears to also be conserved in nature like energy linear momentum is denoted by the letter P with a vector hat over it and it’s the product of inertial mass and the velocity of the object we can write this in calculus notation as the mass times the first derivative of the position Vector with respect to time now there’s another momentum quantity that’s associated with a body that can rotate as well so the degree of its rotation around some axis imparts some angular momentum to the system and we also learn that enclosed and isolated systems this quantity can be conserved it’s proportional not to the mass of the body but to the distribution of mass around the axis of rotation the so-called moment of inertia and to the rotational velocity of that body all points on a rigid body that can rotate about an axis will have the same rotational velocity regardless of their distance from the axis of rotation and the moment of inertia describes using an integral which is shown here I is the integral of r² DM where R is the distance from the axis of rotation for the little bit of DM mass that you’re considering at the time the product of these two things yields the angular momentum and this is observed to be conserved in systems that are closed and isolated now if an external conservative Force acts one where the work done by the force in moving an object from point A to point B is the negative of the work moving from point B to point a by any path that you can take then there is an Associated potential energy as well which we denote U this is another kind of energy so there’s kinetic energy and then for conservative forces where those things like gravity for instance can act on a system you have an Associated potential energy you can lose kinetic energy and store it in potential energy when you can lose potential energy and gain it in kinetic energy there’s an interplay in these kinds of energy in systems and the total energy can be conserved on the other hand for external non-c conserv ative forces such as friction or Air drag there is no Associated potential energy but other forms of energy such as heat which is the motion of atoms in a material object can result from losses of kinetic energy through the action of those forces now as I’ve hinted at before energy and momentum can be conserved and for a system that is acted upon only by conservative forces which have an Associated potential energy and is otherwise close osed to and isolated from all other kinds of forces in that specific case what is known as mechanical energy is completely conserved mechanical energy is the sum of all kinetic and all potential energy in the system at any moment so for instance there might be some initial moment of time where there’s a total kinetic energy Ki and a total potential energy UI and if the system obeys the constraints I’ve listed above that I can look at any other time say some time final later denoted with an f and I can see that although kinetic and potential energies may have morphed one into the other the sum of these two things across all objects in the system is the same sum as I had at the earlier time now for a non-closed and nonisolated system and especially where non-conservative forces can act total energy will be conserved but not just mechanical energy and total energy is the sum of kinetic potential and all other forms of internal energy like heat due to friction or drag or even chemical energy if for instance mechanical energy has been converted into stored chemical energy through some chemical and mechanical and electrical process then you can retain the energy in that form and you may be able to get it back later in the form of either potential or kinetic I energy depending on what kinds of non-conservative forces are acting in the system but if you can figure out all the energy buckets where energy can go in a system even one where non-conservative forces can act then you can still see that the total energy in all of those buckets added up remains constant over time even if you can’t recover mechanical energy when it’s lost into forms like heat or chemical energy and for a closed and isolated system of objects total momentum both kinds linear and angular is conserved so if I sum up all the linear and all the angular momentum at one time initial TI I will find later on that the sum of all momentum and all angular momentum all linear momentum and all angular momentum is the same even if it’s been interchanged between objects maybe they’ve collided with each other things like that now if only elastic Collision of these objects are possible that is the number and mass of the objects never changes then the total momentum and kinetic energy are conserved in that case but if in elastic collisions are possible where objects can stick together for appreciable periods of time or if they can lose mass or gain mass then only momentum will be conserved but again you have to be very careful with how closed and isolated the system is now another law that we encounter in introductory physics which seems a strange Beast compared to the other kinds of mechanical phenomena that we encounter in in these courses is the law of gravitation which governs the gravitational force between any two bodies with mass it acts without physical contact and it does so even across empty space and I’ve Illustrated that here by showing you the planet Jupiter which is the heaviest planet in our solar system and four of its moons the ones that were first spotted by Galileo when he turned his telescope to the night sky to see what he could see these are the so-called Galilean moons they’re the biggest moons of Jupiter Jupiter has many more moons than this but these are the four most visible the most easily visible even with a modest uh Aid to the eye and those are IO Europa ganam and Kalisto and the these four moons do an orbital dance around Jupiter they don’t orbit the earth they orbit this planet and this was a remarkable observation in the days of Galileo that you had objects in the night sky that didn’t go around the earth and they do this under the influence of gravity the same force that holds our moon in orbit around our planet and our planet and all the other planets of the solar system in orbit around the Central Star our sun it’s gravity gravity explains all of this stuff now the gravitational force that an object a exerts on an object B is proportional to the masses of both objects and inversely proportional to the square of the distance between them and this is codified in the law of gravitation that is the gravitational force between any two bodies so for instance the force on a that’s exerted by B is proportional to the product of their masses divided by the distance squared between them the constant of proportionality G I’ll get to in a moment but the force points from the object that’s acted Upon A toward the object that’s doing the acting B so it’s an attractive Force now again this is the force that a experiences exerted by B now G is this Universal constant of proportionality it must be determined by experimental methods and it’s currently known to be about 6.67 * 1011 new m s per kilogram squar not a very big number gravity may seem like a strong force but that’s because we’re being pulled on for instance by all the atoms of the planet Earth and that’s why when we try to jump off the surface of the planet Earth we get pulled back down to the surface so all the atoms of the earth below us are pulling back on us as we attempt to accelerate away and it re accelerates us back to the Center of the Earth but of course we don’t go through the surface of the Earth when we hit it why is that that’s because another set of forces electromagnetism governs the interactions between atoms and atoms tend to repel each other because they have clouds of electrons around them and the electrons have the same electric charge and in the laws of electromagnetism this causes a repulsive Force to occur and so while gravity may seem strong the truth is because we don’t get pulled through the surface of the Earth and down to the core of the planet is because of the strength of electromagnetism which overcomes an entire planet’s worth of atoms pulling on you now what’s worse gravity seems like a strong force but it’s not and also this this Force law doesn’t really tell us its origin it it has something to do with mass and it it w weakens or or strengthens depending on your distance squared between two objects it tells you what direction it points but it doesn’t explain what the origin of gravity actually is what is this Force where does it come from so one of the unsatisfying things about the law of gravitation is that it’s very descriptive but it is by no means explanatory and this was something that even Isaac Newton recognized and because he could provide no evidence to explain the origin of the force known as gravity he preferred not to speculate on it and left it open for the people that would come after him to try to figure out but it was certainly one of those puzzles he never managed to resolve in his lifetime and its resolution would be left until the modern era of physics now speaking of the laws of electricity and magnetism let’s take a look at those and I’m going to do so in a form that may not be very familiar to you but it will be beneficial to you later even if you don’t completely understand the notation now electric and magnetic forces have something in common with gravitation they can act without physical contact across stretches of empty space however it’s pretty much right about there that they part ways from Gravity there strength is proportional to a completely different physical property of nature electric charge which various bits of matter like the electron for instance appear to carry as a fundamental property now like Gravity the strength of say the electric or magnetic force appears to vary inversely with the square distance between charges or flows of charges and depending on the situation we’re talking about here but I can wave my hands sort of make that rough approximation a density of electric charge however is the source of the electric field of force Mass has nothing to do with the electric field of force it has something to do with the gravitational field of force but again this is roughly where gravity and electricity and magnetism all part ways now an electric current density that is a flow of electric charge is the source of a magnetic field of force so a static electric charge just sitting there in space will exert an electric force on another charge somewhere nearby but in order to get a magnetic interaction to occur one of those charges has to be moving relative to the other now I’m going to define a symbol it’s this funny triangular symbol known as nabla because it resembles a ancient harp of the same name it’s got a little Vector sign over it which immediately tells you that whatever this thing is it has directional information and it’s funny because it’s not made of numbers it’s made of derivatives and specifically it’s made with either the uh full or partial derivatives with respect to space so for instance the derivative of something with respect to X the derivative of something with respect to Y and the derivative of something with respect to Z this exposed triplet of derivatives is known as an operator it doesn’t itself return a number but when used on another thing like another Vector it can return a number so you can think of it as a function that when finally given something on the right hand side to act on will give you some information back but on its own it doesn’t really give you information it’s just prepared to tell you how something changes in space now you may not have seen this symbol before and that’s okay but by defining it it allows me to write the laws of electricity and magnetism so-called Maxwell’s equations in four compact mathematical lines now the laws governing these electric and magnetic fields are four in Number the first one is known as gauss’s law for electric fields and believe it or not from this this compact little equation here you can under special conditions derive kul’s law which is probably what you really learned was the law of the electric force in introductory physics there is a simple exercise one can go through to show that this reduces to kul’s law but this is the most preferred in general form of this particular law of electricity and magnetism and in English what it tells me is it tells me that a charge density that is a charge per unit volume row is the source of an electric field on the left hand side we have this operator I defined above which is just a triplet of space derivatives acting on an electric field via the action of the dot product so this thing returns a number and that number is equal to the charge density divided by Epsilon knot which is a constant of nature the second law is gauss’s law for itic fields and this one is probably the simplest of the four it’s that same operator action the nabla symbol with a DOT product with the magnetic field but on the right hand side you get zero and what this equation tells you is that so far as we know there are no such thing as a magnetic charge in order to create a magnetic field you have to have moving electric charge and so far as we know and many experiments have tried and many experiments have failed uh there is no such thing as a magnetic charge that’s what this equation codifies then there’s the Faraday Maxwell law the Faraday Maxwell law tells me that if I have a time-changing magnetic field this can generate an electric field now I have a different Vector operation on the left hand side I have this nabla symbol the vector cross product with the electric field which returns a vector and indeed I have a vector on the right hand side as well the time derivative of a vector field is also a vector and then finally there’s the Ampere Maxwell law and this tells me something a little bit similar to the Faraday Maxwell law and that is that if there’s a time-changing electric field or if there’s a current density of electric charge a flow of electric charge or both then this results in a magnetic field so the left hand side tells me that there’s a magnetic field that exists the right hand side tells me where those magnetic fields might come from either from a charge current density or from a Time varying electric field and mu KN here is another fundamental constant of nature Epsilon KN and mu you should have encountered in introductory physics and you can go ahead and look up their values now what’s amazing about the laws of electricity and magnetism Maxwell’s equations is that when you consider them in a particular situation it finally clarifies what the heck the nature of light is light is an amazing phenomenon it carries information from one place to another and it does so at a seemingly immense speed and it turns out that by solving Maxwell’s equations in a certain regime you find out what light is it’s a very rewarding exercise one that you would presumably go through in a more advanced course than this one but I’ll tease it here so for instance if you consider empty space where there are no electric charges no row no charge densities and where there are no electric currents no JS with the vector hat over the top of it um nonetheless Maxwell’s equations are not just simply all zero so let’s take a look at those equations under those conditions I’ve Rewritten the four equations with no electric charges and no current densities so I have this uh nabla e Vector is zero NAA B Vector is zero I have NAA cross e Vector is just negative dbdt and no across B is something proportional to the time derivative of e so there is a trivial solution to this e and B can be zero that works out just fine but there’s another solution to this that isn’t the so-called trivial solution and the non-trivial solutions are vector functions of space and time and this is what they look like the electric field and the magnetic field as a function of space and time that also satisfy these four equations are these time and space varying functions over here they’re cosinusoidal and they can all be written in terms of the electric field they describe some kind of oscillatory phenomenon oscillatory phenomena like waves are things you should have learned about in an introductory mechanics Class K hat here simply indicates a unit Vector that’s in the direction of travel of the phenomenon and this number c with a zero subscript that turns out to be the speed of the phenomenon in empty space because that’s the kind of space we’re considering here empty no matter no charges no currents and it turns out that you can solve for that speed and you find out that it’s equal to 1 over the square root of those fundamental constants of nature mu * Epsilon KN and if you plug those numbers in you get an amazing fact out of this that whatever this phenomenon is it travels at 2998 * 108 m/s and for the astute among you this is the speed of light so what Maxwell’s equations in empty space tell us is that when solved they describe a phenomenon that can travel from point A to point B seemingly through empty space and it does so at precisely the speed at which light was known to travel in the days when this was solved so light is what is known as an electromagnetic wave and like a mechanical wave which was the only analogy that physicists had at the time it was originally assumed that it must travel in a medium sound travels in air water waves travel in water they are distortions of a medium and so it was presumed that light must too be some kind of mechanical wave and that means that seemingly empty space couldn’t really be empty something’s got to be there that distorts to allow this wave to travel that was the Assumption based on mechanics now finally I want to go into the subject of Relativity which would have been introduced to you probably under the phrase relative motion in introductory physics you get some exposure to relative motion that is a person standing on a train the train is moving relative to somebody on the ground the person on the train throws a ball up in the air what do the person on the ground see that’s usually the way in which this is couched the person on the train for instance who throws the ball straight up in the air will see it go up gravity will accelerate it and eventually it will come straight back down into their hand so it just goes up slows to a stop and then accelerates down back to their hand all along a straight vertical line that’s what the person on the train sees a person on the ground watching this sees the ball follow a parabolic trajectory because the ball and the person have a horizontal velocity because they’re standing on the train so the ball goes up and comes down yes but it doesn’t land at the same coordinate along the horizontal that it started at it appears to follow a parabola and so the two observers will disagree on the motion of the ball the person on the train says no no no it goes straight up and then comes back down to my hand and the person on the on the ground says well no it didn’t go straight up it followed a parabolic trajectory but your hand moved too and so it was there to catch it when it came back down and it’s possible to use mathematics to relate these differ observations of space and time uh and to do this you assume that time passes the same for all observers the person on the train and the person on the ground all experience time the same way and when you make that assumption you get out of this something known as as the Galilean transformation that allows you to relate spatial coordinates and velocities of objects from a frame you consider to be at rest to a frame that you consider to be moving so in our case you might consider the platform or the ground next to the train to be the rest frame you might consider the train to be the moving frame and these equations shown down here will relate coordinates velocities and times in the moving frame with the primes next to them to things in the rest frame the numbers without the primes attached to them okay so that’s not so bad it’s actually one of the more complicated things that most students encounter in introductory physics because it forces you to think in two different frames of reference and this is not always as straightforward as it seems but the math itself is not that bad it’s more the conceptual issues that go along with this that that pose a particular challenge for most people who see this the first time so that is basically a summary of of what we now call classical physics introductory mechanics and the laws of electricity and magnetism or semester 1 and semester 2 physics and even though classical physics is challenging there are many difficult things that you have to do there’s new math you haven’t seen before you’re often learning calculus at the same time you’re expected to use calculus in introductory physics nonetheless at the end of the day if you stop and look at all of this stuff you’ll often say okay the mathematical or some conceptual difficulties side all of the stuff feels to me very intuitive I can throw a ball up in the air I can catch it I can watch somebody do that in a train and see it moves in a parabolic Arc okay yeah we disagree on on what’s happening but we can explain to each other why we see what we see it’s all very you know normal day-to-day human scale stuff really this is intuitive it just had to be described by mathematics and that that often is the difficult part but you have to be very careful about intuition intuition is largely based on experience with events that involve the following things speeds that turn out to be very close to zero you know driving at 70 M an hour may seem really fast to you as a human being or getting on a rocket ship that goes into Earth orbit might seem really extreme and they are for human beings but compared to the fastest known phenomenon in the universe which is light 2998 x 108 m/s 70 M an hour seems pretty pathetically slow and in fact is so close to zero that from the perspective of light it might as well be nearly at rest not very impressive to light so you have to be careful one because the speeds that you’re used to encountering are really close to it turns out zero and so your intuition is built on a very narrow spectrum of experience in the universe the other thing that you may take for granted is that the sizes of things that we usually think about In classical physics with the exception of electrons and protons and electricity and magnetism the sizes of those things tend to be very large by comparison to what are known to be the building blocks of the material universe and for the stuff around us that’s mostly going to be atoms that’s the day-to-day stuff that we are interacting with but when you interact with a table that table has like avagadro number worth of atoms in it that is a huge number of atoms and the scale of the structure built from those atoms is vast by comparison to the atoms themselves and so as a result as we begin to encounter phenomena and this was true of physicists at the end of the 1800s as you begin to encounter phenomena that are very fast or very small so objects moving very close to the speed of light or objects that are really more at the atomic or even the subatomic scale the things that make up the atoms you begin to find that classical physics needs to be be modified to describe the universe more completely it works for slow things at large scales like human scales or plane size scales or even bigger but it breaks down in regimes where it was never designed to operate the very fast and the very small so as a result you’re often going to find as you go into modern physics that what you think to be true about the universe is based on intuition from a limited set of experiences in the cosmos and as a result your intuition is actually fundamentally wrong but the good news is is that this only means that you are finally finally experiencing the breadth of the universe all it has to offer at all of its scales in speed and size rather than that limited scale of phenomena closer to Human Experience so let’s use classical physics and let’s make some predictions to set ourselves up for where people started to go really wrong with these ideas in roughly the late 1800s now the tenets of classical physics which I can summarize based on the earlier part of this lecture are encoded largely in Newton’s laws and Maxwell’s equations and they should if this is all there is to the universe apply to all phenomena in the natural world after all if this was really the complete set of all the laws of nature that had been disced discovered in the 16 and 1700s that it must be true that they describe everything otherwise they’re not a complete set of laws so let’s take a look at light what would the framework of classical physics then insist be true about light well from Maxwell’s equations we know that light is some kind of oscillatory phenomenon like a wave and so our experience with waves in the 1800s was that they must be mechanical in nature they must represent the Distortion of a medium so they gave it a name they named it before they ever discovered it and they called it The Ether and it was believed to be the thing that actually fills empty space empty space isn’t empty it’s made of this substance called The Ether that we normally can’t experience but light experiences it and the Distortion of the ether is what we call light that was the hypothesis based on the mechanical understanding of wave phenomena so the speed of light in so called empty space the number that we got from Maxwell’s equations that isn’t really the speed of light in empty space it’s the speed of light measured relative to an observer at rest with respect to the ether The Ether is the universal reference frame for light and if you can be at rest with respect to the ether then you will observe that light moves at 2998 * 10 8 m/s it’s a big number okay but this would then make ether the universal rest frame that is the the frame that you could Define to always and absolutely be at rest and then everything else is in motion relative to it that would be awesome the Galilean relativistic and Newtonian mechanical view of the universe would have allowed something like this to exist now the problem was that sort of the new kid on the Block Maxwell’s equations which really only emerged in the you know second half of the 19th century they were silent on the topic of The Ether they described no substance that required this electromagnetic wave called light to propagate so it was assumed that they must be incomplete that the new kit on the Block they’re probably not complete they need to be completed and The Ether would complete them so it was assumed that Newton’s mechanical view of the universe the laws of motion and all that stuff that that was correct but that Maxwell’s equations was just incomplete and needed needed to be completed with this mechanical substance The Ether so if we then apply this thinking to a problem involving light and travel and time what would we predict let’s put ourselves in the role of sort of late 19th century physicists we’ve learned all this stuff it’s been solid for 200 years so what are we going to predict so let’s do a thought experiment a thought experiment is a kind of experiment that you can carry out entirely inside of your head what you do is you imagine a scenario you analyze the scenario using the understood principles of nature or laws of physics and you look to see if the conclusions of running this imaginary experiment would in any way violate logical or physical consistency and if you determine that that’s the case you may have hit upon a useful inconsistency in our our understanding of nature that could then be used to figure out what the correct description of nature might be so to do our thought experiment let’s imagine that we are in a space that is filled with ether the medium in which light traveling as a wave disturbs the medium and propagates at 2998 * 108 m/s now imagine into this volume of ether we place two cars one car at the left one car at the right and the car at the right has its headlights aimed at the car on the left so that an observer in the car on the left could look back out the window and if the headlights of the car behind them were on they should be able to see the light but let’s put a 30 km gap between the front of the right car and the back of the left car so that light if it wants to go from the car on the right to the car on the left has to cross this gap of 30 km okay fine so we’ve placed the cars in The Ether the cars are at rest with respect to the ether so they’re in the frame of reference of The Ether and the car on the right switches on its headlights how long does it take for an observer in the car on the left the second car to see the light reach them well this seems pretty straightforward right you know the distance it’s 30 km from where the light leaves the right hand car and arrives at the left-hand car and Maxwell’s equations tells us that light travels at a fixed speed it doesn’t say anything about The Ether but we’ve invented The Ether to help us to have electromagnetic waves comport with all prior knowledge of mechanical waves so it’s a medium with mechanical properties that can stretch and squash and those stretchings and squashing are electromagnetic waves and in that medium light travels at 2998 * 10 8 m/s okay everything’s at rest with respect to the ether light travels at the speed of light in ether so we just run the numbers we take the distance we divide by the speed and we get the time that is required to make this journey and we find that that time comes out to be about .1 milliseconds 1 * 10-4 seconds okay nothing hugely revelatory here but let’s take our thought experiment One More Level forward now let’s imagine that both cars have been plopped into this ether volume and they accelerate at the same time up to a constant velocity of half the speed of light that’s a speed of 1.5 * 108 m/s and let’s imagine that the cars are both moving together at the same velocity from right to left so they’re traveling from the right to the left in The Ether at all times they maintain a fixed distance between the front end of the right car and the Observer at the back end of the left car of exactly 30 km the car on the right turns on its headlights now how long does it take the light to reach the observer in the other car well let’s review what we think we know about light speed and this so-called ether that distorts to allow electromagnetic waves to propagate light travels at C the number given by Maxwell’s equations 2998 * 10 8 m/s in the rest frame of The Ether but now from the perspective of the cars the ether is a wind that’s rushing past them still air on a calm day leaves no sensation on your body but if you were to start running forward you would perceive a wind hitting you in the face and that’s sort of the equivalent situation here both of these cars are now traveling through the ether they’re doing so at half the speed of light and so from their perspective The Ether is rushing past them as a wind and its speed is also half the speed of light it’s as if they perceive themselves to be at rest and The Ether to be rushing past them at half the speed of light so the velocity of this wind is the negative of their velocity with respect to the ether now Gile relativity and Newtonian mechanics demand that from the perspective of observers in the car that the light that leaves the car on the right while it’s traveling at 2.99 x 108 m/s in the rest frame of the ether is encountering this wind of ether that has the apparent effect of slowing it down this is sort of like sound waves or water waves in their respective media if the medium is moving then the medium’s speed can add or subtract from the velocity of the wave in that medium and so Galilean relativity and Newtonian mechanics are going to demand that the observed speed of light in the frame of the cars is the speed of light in the rest frame of The Ether minus the velocity of the cars and so you would actually see the light leaving from the right hand car and traveling the gap between the right hand car and the left-hand car and what seems like a slowed speed as if it’s encountering resistance as it moves forward it’s not moving at 2998 m/s anymore it’s moving at about half that and so you would answer that well the distance between the cars is still the same it’s 30 km and the speed of light has been reduced by The Ether wind and so you you would predict based on all knowledge at this stage that the time it takes for the light to get to the other car is greater than it was before it’s about2 milliseconds now twice the time that was required when the cars were at rest with respect to the ether now that’s a prediction and it comports with all prior experience in the pre-20th century World it comports with ideas about how velocities add in relative motion it comports with the idea that waves can only travel because they distortions in some kind of medium a mechanical explanation for waves that’s consistent with Newton’s mechanics all of this seems to be perfectly acceptable from the perspective of the Bare Bones introductory physics to which you would have been exposed but a fair question to ask is this is the outcome of a thought experiment what would be observed in a real experiment in the real world and we’ll take a look at that so let’s review the basic ideas that are the foundations for modern physics the groundwork for modern physics are Newton’s mechanics the concepts of energy and momentum quantities associated with motion that can be conserved under certain conditions the law of gravitation and the laws of electromagnetism however these were largely built to describe phenomena that comport with typical human experiences phenomena at our size scales or slightly larger or smaller essentially within our ability to see the world around us including with a mic microscope or a telescope that would all be within the human scale um the exception however is Maxwell’s equations they were developed by studying electric charges which are very small and they are really beyond the scale of everyday experience except in their large scale macroscopic effects like electric and magnetic forces electric currents lightning strikes refrigerator magnets things like that they have these big macroscopic effects that feel familiar to us but but at the individual level of a an electron let’s say things are not typical compared to the human world by the end of the 1800s chemists and physicists were beginning to directly interact with scales that really were Beyond Human Experience so for example the electron is discovered in 1897 and it turns out to be the first subatomic particle although that really wouldn’t be fully understood for several more decades in addition an invisible radiation like for instance what we now call x-rays this was discovered at the end of the 1800s and 1895 in the case of X-rays and these phenomena and other phenomena at the same scale even atoms themselves or other General forms of light they turn out to be Way Beyond Human Experience and so trying to adapt our intuition in the form of Newton’s Mechanics for instance to these phenomena would lead to spectacular fails now not only were such new phenomena small they also turned out to be capable of moving extremely fast x-rays move at the speed of light electrons with minimal effort can be compelled to move at almost the speed of light such speeds are also very much Beyond human day-to-day experience although you might lead yourself foolishly to think that you understand them really well so this concludes a foundational lecture a review of the material you should have been exposed to already in semester 1 and semester 2 physics I know that I’ve couched this in some ways that are unfamiliar but I’m trying to Rattle you out of any complacency you might be in after having had a couple of introductory semesters of physics and we’re going to begin to explore the consequences of these classical physics predictions on phenomena like light in class and then we will build on what we conclude from those Explorations into the first steps of modern phys [Music] physics in this lecture we will learn the transition in thinking that led from Galilean relativity to the special theory of relativity in 1905 we will learn the postulates of special relativity which are the basis of the mathematics of the framework and we will look at some of the consequences of those postulates even before we delve into the mathematical framework itself in class we looked at the lessons of the Michaelson Morley experiment which can be summarized ized as follows first light travels at a fixed and constant speed in any medium regardless of the relative velocity of the light source and the light Observer this is unlike any other phenomenon described in mechanics and it implies that Newton’s mechanics is actually the incomplete theory of nature no medium is actually required for that light to propagate unlike a mechanical oscillatory phenomenon a wave to exist light requires no medium to be distorted it is not mechanical in origin and this implies that Maxwell’s equations are complete or at least sufficiently complete to understand light these lessons however would not be fully absorbed until about 1905 when Albert Einstein one of the most famous physicists in history published the definitive papers explaining how to reconcile mechanics electricity and magnetism and the results of the Michaelson Morley experiment now interestingly the mathematics that Einstein would come to rely on for encoding the relationship between space and time measurements in one frame and space and time measurements in another frame were actually laid down much earlier by Hendrick Loren in a famous paper on the compression of bodies in The Ether the mathematics that would later become a replacement for the Galilean relativity equations would actually be kind of sketched out but for a completely different purpose than they would ultimately be used for Loren was considering the effects of The Ether on bodies that are moving through it now these bodies are held together by chemical bonds they’re made of atoms and those atoms are chemically bonded to each other but chemical bonds are just electromagnetism in action and So based on this he arrived at a few hypotheses should The Ether exist first that mechanical bodies would compress along the direction of motion in The Ether and this has a precise mathematical description for the process and second in transforming observations from The Ether frame to other frames of reference he would conceive of an alteration of time that also had a very firm mathematical description now Loren conceived of this during a period when The Ether was still very much believed to exist the results of the Michaelson Morley experiment were not fully digested during this period the ether’s existence would ultimately be disproven or at least shown not to be necessary to explain anything that was then known about nature in the decades that would follow this work but the mathematics laid down by Loren during this period would still prove extremely useful and today we know this as the Loren’s transformation the replacement of the Galilean transformation from frame to frame we’ll come back to that in a later lecture let’s talk about Albert Einstein and his miracle year of 1905 so Albert Einstein in 1905 was a young PhD physicist who was laboring doing physics as sort of side work in what was otherwise supposed to be his regular work at the Swiss patent office in Burn Switzerland he had this job because he was unable to secure for instance a faculty job after completing his PhD and in part this was because Einstein really couldn’t get any recommendations out of any of the professors that had supervised his education because he had so irritated them with his behavior during what we would consider graduate school including skipping out entirely on classes uh in particular for instance math classes for mathematics he didn’t consider to be physically useful uh and also for challenging his professors challenging their Authority thinking of them as idiots and so forth now Einstein was a very bright young man but he was also a bit arrogant and temperamental and this didn’t do him any favors when he was trying to get a job now ultimately it was the thinking that culminated at the end of his PhD work and then into the years leading up to 1905 that would lead to a change in the way that the community of physicists thought about the supremacy of the assumptions made in Newton’s mechanics versus what the laws of electromagnetism that is Maxwell’s equations had to say about light and space and time and in 1905 he published the work that had resulted from his PhD research in a series of about four papers and this was his so-called miracle year this is a highly productive year for a young and relatively unknown physicist in this day in doing so he reframed assumptions about space and time and what is and what is not invariant to all observers and all frames of reference recall that in the neonian and Galilean view of

space and time time is experienced the same way by all observers regardless of their relative states of motion time would be referred to then as an invariant but what Einstein proposed challenged thinking about what was and what was not invariant in space and time now it in short here’s what Einstein did he accepted the conclusion of the Michaelson Morley experiments that light has a fixed speed regardless of the motion of the source relative to The Observer of the light from the source this then implied that there’s no ether as well using a simple thought experiment like the one that we did in the foundations lecture involving car headlights and The Ether he explained also why time is not absolute even in Newton’s mechanics time itself self is not an invariant concept and he did a quick thought experiment that showed that it wasn’t even true under Newton’s way of thinking and so he was free to abandon time as the constant thing in Transformations from one frame to another frame instead he chose to preserve overall the forms of the laws of physics and the speed of light which the Michaelson morly experiment implied was constant regardless of your state of motion this then led to the foundation of two postulates that allowed him to then Define all the mathematics that would follow the first postulate is what I hinted at a moment ago the forms of the laws of physics that is f equals ma for instance or Maxwell’s equations will be the same for all observers regardless of their state of relative motion that is their frame of reference the second postulate is that the speed of light is the same for all observers regardless of their frame of reference their state of relative motion now let’s begin by breaking down the concepts that we need to dig into so that we can really understand where all of this is headed we need to take these postulates and parse them into some phrases and words Define those things and then go forward from there this will allow us to build back up to a more complete understanding of the math that we’ll eventually need in order to understand relativity and relative motion going forward in modern physics first off there’s the word event you might think you know what this is but in physics it is given a very precise definition so that we can always try to define the concept mathematically so that everyone can agree on what an event is and what an event is not another phrase that’s deceptive and may seem to have a common definition for you but where we have to be careful about this in physics is the phrase frame of reference we need to Define it it comes up a lot in our discussions and because descriptions of events can depend on the frame of reference in which the observation is made we have to carefully Define this concept simultaneity is another word probably the one that causes the most consternation among people who are making the transition from introductory mechanics and electricity and magnetism into modern physics because simultaneity has probably been implied in a lot of things in the past but we have to put it on some firm footing conceptually here so that we can use it and explore it going forward it turns out that the concept of simultaneity is actually essential to many things you take for granted all the time you’ve just never been forced to think about it before this concept turns out to be a subset of the discussion of events and it’s going to play a very important role so we’re going to have to Define this and then finally you might think you’re comfortable with this idea but the phrase speed of light would benefit from some context and some description we should really try to understand the number that is behind this phrase It’s a ridiculously large number compared to most things on the day-to-day human scale but it actually turns out that this speed is only impressive on the scale of things that are roughly the size of planets I’ll even allow solar systems uh and maybe smaller depending on how you define a solar system but it turns out the speed of light is not as fast as we would like it to be um and certainly on the scale of things like the entirety of the universe it is pathetically slow so let’s get started in the next few slides trying to Define each of these things very carefully first of all let’s talk about the concept of an event an event is quite simply anything with a location in space and time so let’s practice this concept I will show you an event and I want you to try to describe it with words and numbers go ahead and pause the video while you’re doing this when prompted see you can come up with a short sentence that describes the event using the definition that an event is anything with a location in space and time this is excellent practice for defining events in any new situation that you will encounter as a exercise in setting up a problem for eventual solving I’ve given you a one-dimensional axis so an X AIS and let’s say that the numbers here have units of meters that’ll make it easy for us to very precisely describe an event I’ve also given you a timer the timer is capable of ticking out about 12 seconds and the units on each of these tick marks 1 two 3 and so forth are seconds okay so you have a spatial reference and you have a time reference given that information let’s go ahead and proceed with looking at an event and attempting to describe it I want you to describe the event depicted above on the x-axis go ahead and pause the video come up with a short sentence that uses the definition of an event to describe it and then resume the video when you’re ready to compare to my answer you should have come up with something like the following the dot is at position xal 0 m at time tal 0 seconds that’s an example of describing an event in physics the dot is at a spatial location that is defined at a time that is also defined x and t space and time locations if you didn’t feel comfortable doing this now that you’ve seen me go through it once let’s try another event you try to describe it and let’s see what you come up with describe the event depicted above now on the x-axis go ahead and pause the video write down a short sentence that uses the definition of an event to describe this event and then resume the the video when you’re ready to see what I came up with so what I decided to do is to describe this as follows the dot is at position x = 2 m at time T = 2 seconds make sure to check your space reference and your time reference when presented with an event so that you correctly Mark in say x and t or XYZ and T the coordinates of an event an event is something that has well- defined coordinates in space and time a location in space and time let’s now talk about a frame of reference a frame of reference is any object or system all of whose Parts move at the same velocity with respect to an agreed upon reference point in space that’s quite a mouthful let’s go ahead and illustrate this with an example I want you to consider the three objects shown below labeled black a black dot blue a blue dot and red a red dot now one of them the black dot is agreed upon by the others the red and blue dots as the common reference point for all measurements now as I’ve depicted them here blue and red have an Associated velocity Vector shown here and as depicted red and blue are in the same frame of reference because they have the same velocities let’s check that if I roughly eyeball the length of this Vector it seems to be pretty similar to the length of this Vector so from this I could conclude that very likely blue and red have the same speed with respect to black but velocity is not just speed it’s not just the magnitude it’s also the direction and here I see that the directions align they Point parallel to each other and so I conclude from this that they have the same velocities therefore although blue and red are both moving they are moving in the same way with the same velocity they have the same state of motion and therefore they are in the same frame of reference now take a look at this one I’ve changed something here does this change alter the conclusion about blue blue and red do the Red Dot and the blue dot share the same or different frames of reference pause the video here look carefully at the image and then resume the video when you’re ready to hear the answer the answer is that they do not although their speeds are the same the lengths of those two arrows look pretty much identical the direction of the mo notion of the Red Dot relative to the Blue Dot and all measured with respect to the black dot has changed this means that they have different velocities and different velocities means different states of motion and therefore different frames of reference have now emerged here the blue frame of reference is no longer the same as the red frame of reference now I want you to consider the objects in this picture blue red and now a purple dot all of their velocities are measured with respect to the black dot as the reference point that hasn’t changed I want you to practice a little bit more and I want you to think about how many unique frames of reference you can identify in the above picture go ahead and pause the video here I’m not going to provide the answer here because I really want you to try to step out on a limb on this one but feel free to talk to me as the instructor outside of class or in class if you’re not confident in how to determine the answer to this question now let’s visit the concept of simultaneity simultaneity is a subset of events in which two events or more are said to be simultaneous that is to possess of this quality simultaneity if they are observed to occur at the same moment in time this seemingly straightforward definition of the concept should not fool you you have to think really hard about whether events are actually simultaneous and if there are multiple observers in different frames of reference for whom are those events simultaneous finally let’s look at the speed of light and let me be clear about the speed of light light it is the number of meters that light can travel once it’s been emitted by some kind of source in a certain amount of time that’s just the old definition of speed but light is special it’s special because the Michaelson Morley experiment tells us that no matter the state of motion of the Observer or the emitter of the light all parties will agree that when they measure the speed of that phenomenon in any frame of reference it always comes out to be the same number 2998 * 108 me per second at least an empty space now the history of the speed of light is interesting it can be cherry-picked through to take a look at what people try to do to measure the speed of this phenomenon because it is ridiculously fast now Galileo galile famously claims to have attempted to measure the speed of light by uncovering a lantern having an assistant on a distant Hill who in response to seeing the light from Galileo’s Lantern then uncovers one of their own and then G Alo upon seeing the assistance Lantern light records the time for the round trip taking into account human reaction time it turns out of course that light moves way too fast for this to work with 17th century technology even if Galileo used the most precise clocks of his day which he had invented water clocks there’s no way that even given 40 50 or 60 miles of distance between him and his assistant that that technology would have been sufficient especially with really slow human reaction times to in fact measure the speed of light so this was kind of a lost cause but a clever technique nonetheless and one which can successfully be used to measure the speed of sound another important person in the story of the measurement of the speed of light is Ol RoR now he would go on to use the period that is the time it takes to complete one cycle of Jupiter’s moon a which had been discovered by Galileo using the telescope and and by looking at its cycle of eclipses by Jupiter to then make the first reasonable determination that light travels in finite time he did this in about 1676 revisiting his data in a modern context suggests he shouldn’t have been as accurate as he was in measuring the speed of light but he actually got fairly close to the currently accepted value certainly impressive for its time uh impressively close to the currently accepted value of the speed of light but one could definitively conclude from his work that light does not travel instantaneously from place to place rather it takes a finite amount of time to cross space even if it does so very quickly now by the time of Albert Einstein’s Publications the speed of light had been established by multiple experimental methods to be within about 50 km/s of the Precision of today’s methods and that is is remarkable for such a large number representing such a incredibly high speed so let’s then take a look at the Modern speed of light and the number that is the currently accepted calibrated value of this speed today and I say that because the definition of things like the meter are based on the distance that light travels in a certain amount of time so based on the current definition of the meter and the second the speed of light is defined to be exactly 299,792,458 m/s or about 2998 * 10 to8 m/s A good rule of thumb something that will Aid you whether you’re thinking about how long signals will take to propagate in electronics or if you’re thinking about how long it will take for a light signal to propagate across some space for a communication system or something like that A good rule of thumb is that light travels roughly 1 foot in one billionth of a second that it goes that is it goes one foot per nanc that’s a handy little thing to remember for engineering purposes going forward now let’s begin to look at the consequences of the postulates of special relativ and I say special because there’s a more general theory of relativity a more general theory of space and time that Einstein would spend another decade working out after 1905 what makes the early theory of space and time that he developed special is that it focused on what are called inertial frames of reference those in which there are no net unbalanced forces now that doesn’t mean that accelerations can’t be present but it is a special case of a more general theory of reference frames space and time now under this special condition an object in motion will appear to all observers in all frames to have a constant velocity even if observers in different frames disagree on the magnitude and direction of the vector so let’s recall his postulates again in light of this special condition for the frames of reference that we’re talking about here postulate one is that the forms of the laws of physics are the same for all observers regardless of their state of relative motion that is regardless of the frame of reference in which they find themselves we’ve looked at the definition of the terminology frame of reference the second postulate is that the speed of light is the same for all observers regardless of of their frame of reference all observers no matter their relative state of motion when they measure the propagation speed of light signals will always find and this is based on experimental observation that light travels at the same speed in every frame of reference even if that frame of reference is moving with respect to the source of the light this is taken to be the thing that is invariant from one frame of reference to another frame of reference not time the speed of light now let’s look at some of the consequences from these postulates starting with the first postulate so the consequences of the first postulate are both straightforward and a little surprising so one of the conclusions you can draw from the first postulate assuming that it’s true is that all physical laws like Newton’s laws or Maxwell’s equations will all all have the same observed form in all inertial reference frames now this is pretty helpful actually because what it means is that regardless of our relative states of motion the basic laws of physics that we can uncover by doing experiments observations of the natural world are not dependent on your current state of motion the moon goes around the earth so from our perspective the moon appears to be moving but the law of gravity has been tested on the Moon by dropping objects there we see no difference between the law of gravity on the moon and the law of gravity on the earth despite the fact that we are definitely in relative motion to one another this has been tested more precisely than just dropping things on the moon but the basic conclusion is that this postulate holds and as a consequence of that basic laws of physics can be determined regard regardless of what your state of motion actually is but this consequence has a flip side it’s impossible based on determining the laws of physics by making observations in your frame of reference to determine whether or not you are actually in motion the analogy I like to make for this one is is being a little sleepy on a train if you’ve ever been on a light rail car or a real passenger train you’ve been a little tired you’re sitting on the car waiting at the station for the train to leave and another train is parked next to you you might doze off for a moment while sitting there looking at the other train and then you might wake up and during the time when you were slightly unconscious your train began to move with ever so slight an acceleration you started to gain some velocity and so when you wake up you’ve missed the fact that there was an acceleration in your frame of reference that caused you to start moving and you might look out the window and see the train next to you moving past you and draw the conclusion that the other train is pulling out of the station you conclude therefore that you’re in you’re in the rest frame with respect to the Earth your train is standing still because you feel no forces and the train next to you is moving but then suddenly you reach the end of the train next to you and you realize that your train is the one moving with respect to the ground and that other train was sitting still the whole time you had no way of knowing that you were actually the frame in motion with respect to the Earth because there were no cues and there’s no experiment you could have done in that 30 seconds while you’re passing the other train that would have definitively told you you were moving and the other train was not or that the other train was moving and you were not and that’s one of the consequences of the first postulate there’s no way to measure even the most fundamental statements about nature the laws of nature and figure out that you are moving and not something else so as a result of this postulate it has to be concluded that there is no such thing as an absolute state of rest or an absolute state of motion all motion is relative all Motion in nature is relative to a reference point you have to pick what that reference point is and depending which one you pick may change the degree of your state of motion or the state of motion of the other frame of reference all motion is relative as a consequence of this postulate there is no experiment that could be done if this postulate holds forever that would tell you that you were moving and something else wasn’t or vice versa now let’s look at the consequences of the second postulate the speed of light is the same for all observers regardless of their frame of reference now the consequences of the second postulate are typically more surprising to a general audience of individuals who start really thinking about this for the first time on their own and these conclusions tend to put most people well outside the comfort zone of typical Human Experience so let’s take a look at these so all observers agree that light moves at a fixed speed this is a singular and variant independent of states of relative motion now that’s already a bit freaky in the sense that you could be driving in a car at 70 M an hour and switch on your headlights and somebody on the side of the road standing still with respect to the Earth measures the speed with which the light from your headlights passes them and they measure 2.99 * 108 m/s exactly not 70 M hour faster exactly the speed that it would travel in empty space if it were emitted from rest and you in your frame of reference could get out on your hood and do some very careful experiment to measure how fast light is moving when it’s emitted from your headlamps and you would draw the same conclusion that the speed of the light is exactly that number from a few slides ago even though uh the person on the ground sees you and the source of your light is moving they still measure the same speed of light you measure that is freaky that some how light is immune to a state of motion of the emitting Source but that is an observational fact it may be freaky but it’s also reality and that means you need to rethink the universe at a fundamental level particularly rethink space and rethink time and so as a consequence of this observational fact of nature the belief that humans typically hold that say time or space or both are experienced in the same way by observers in different states of motion has to be completely abandoned if we are to hold the speed of light constant in all frames of reference you have to abandon the absolute nature of for instance time time passing will be experienced differently by observers in different frames of reference so as a result of this postulate there’s just no such thing as an absolute measure of time or an absolute measure of space I mean we already have to abandon the notion of an absolute frame of reference in space from the first postulate but in the second postulate we also find that we can’t hold on to this seemingly intuitive belief that time passes at the same rate for all people regardless of their state of motion measurements in one frame of reference regarding space and time distances need not agree with measurements in a different frame of reference but all observers will agree that light signals travel at a specific and fixed speed independent of the relative states of motion so really special relativity is not so much a theory of what is relative it’s a theory of what is invariant between observers in different frames of reference and it allows us to define a mathematical framework to figure out how to relate our observations so let’s take a look at the relative nature of time briefly using a variation on Einstein’s thought experiment or as he called them gunan experiments gunan from the German for thoughtful or mindful what freed Einstein to write down the postulates ultimately of what we now call special relativity was his ability to be able to abandon Newton’s old idea of absolute time that is time that passes the same way for all observers regardless of their state of motion it was this thing that was really a key moment for Einstein of insight a moment when he he relates that the damn kind of broke at that moment in his mind and he was freed to draw the conclusions that ultimately LED down the correct path to the correct description of nature so let’s take a look at a variation on the gonan experiment that he felt liberated him from the sort of tyranny of absolute time that had been passing down through the generations as an assumption that turned out not to be true so while riding on a street car in Burn Switzerland where he worked as a patent clerk Einstein began to think more carefully about what it meant to know the time by observing the Clock Tower so shown here on the right hand side of the slide is a picture of the burn Clock Tower and here you can see the the tram lines in the street that likely carried the street car on which he was writing at the time when he finally had one of his moments of insight into this question what does it mean to know the time by observing the clock tower well we’re going to do a modern version of his thought experiment because analog clocks are not as common as they were in his day time is the measure of distance if you want to think about it that that way between events that occur for example at the same spatial coordinates so imagine not a analog clock on the face of this clock tower but rather a large blinking light and when the light is on that marks a moment of time it’s an event it has a location in space and a location in time and then the light goes out and then it comes back on later in the same position in Space the gap between the two blinks is what we refer to as a duration of time and we could use that gap between these regular blinks of the light to define a standard unit of time whatever we choose that to be the second for instance now Einstein realized that the way you know that time is passing is you see these two events but to see these two events you need to receive light from the blinking light and light has to travel through space so if you’re on the street car and the street car is moving away from the clock tower the light from the clock tower has to travel from the tower to your eyes so you see the blink after it’s actually occurred but in your frame of reference in the street car it’s the arrival of the light that tells you that a moment in time an event has occurred and then you wait for the next blink to occur but by then the street cars moved a little further away so the light has to travel a little bit further and that takes a little bit longer and so in your frame of reference in the street car time appears to be slowing down and this is just using a Newtonian view of the Universe I haven’t even invoked the postulates of special relativity here this is just the simple fact that light has to travel across a distance and it does so in finite time and the time intervals are stretched by moving away from the clock tower because light has to catch up to you so even in Newton’s view of the universe time measurements cannot be absolute as a result of this so imagine two observers that are using a blinking light to measure time they agree that the blinking of this light is how they will Define their Standard Time units now one of The Observers is at rest on the ground with respect to the source of the light maybe standing right next to the blinking light and the other is on a super Train That’s racing away from the light source and it’s doing so at a ridiculous speed half the speed of light so the two observers agree to count how many blinks occur while the super train makes a journey of 2 million miles now I chose that because this is about how far light can travel in 10 seconds now on the ground The Observer at rest with respect to the blinking light counts 10 blinks during the journey each blink being 1 second apart but for the observer in motion not all of those 10 blinks will have had time to reach the super Trin by the time it arrives at its agreed upon destination it will have marked off fewer observed blinks from the light and thus an observer on the train would rightly claim that less time was required than the 10 blinks that the person on the ground saw to make the journey two observers disagree on how much time has passed using a common reference point so even in Newton’s view of space and time the notion of an absolute time measurement is just not correct now this thought experiment is essentially based on an optical effect you could even say it’s based on an optical illusion the transit time of light through space but nonetheless because it already using a Newtonian view of the universe disproves that there is such a thing as a notion of an absolute unit of time that passes the same way for everybody this completely then frees a thinker from abandoning the concept of absolute time as a necessary tenant of reality so the speed of light is the same for all observers regardless of their frame of reference and since space and time displacements are not experienced the same in frames with different relative states of motion even based on this optical illusion based thought experiment observers at rest looking at an object in a frame of reference that’s moving with respect to them will observe that that object is contracted in length along the direction of motion now we will firmly see that when we explore the Loren transformation for relating observations in one frame to observations in another frame but already you could have concluded that since it’s the speed of light that remains fixed not time or something else that you’re going to have to give something in this process and what you find out from all of this is that objects in motion from the perspective of people who are in the frame that’s agreed upon as being the rest frame will be observed to shorten along their direction of motion this is known as length contraction so hang on to that phrase because it will come up over and over and over again it refers to this phenomenon of spatial measurements from the perspective of observers at rest looking at the moving frame getting contracted in the moving frame now observers in motion relative to other observers will will also experience a slower passage of time it’s not an optical illusion that you need to use to explain this it’s a physical change in the experience of time itself no Optics required to explain the phenomenon it simply is a behavior of time that for objects in motion relative to other observers if they stop moving and then compare their clocks to people on the ground they’ll find out that they have experienced less passage of time than their colleagues who remained in what was agreed upon to be the original rest from frame this is known as time dilation that time slows down in a moving reference frame relative to a frame that’s at rest now it will be a lot easier to appreciate the degree of these consequences as we actually explore the postulates of Relativity in class and then in the next section of this class look directly at the Loren transformation which is the correct way to relate observations between frames of reference so so I want you to get these Notions of terminology down you don’t necessarily have to agree that this is what happens right now because I have done no math to prove to you that this is possible and I’ve certainly shown you no experimental results to tell you that this is what happens but for now look at the terminology and understand what a length contraction or a time dilation is so that we can carry that terminology forward with us so to review what we have done in this lecture we have learned about the following things we’ve learned about the transition in thinking that led from Galilean relativity to the special theory of relativity in 1905 we’ve learned about the postulates of special relativity which are the basis of the mathematics of the framework and further we’ve started looking at the consequences of those postulates from the fact that it’s impossible to tell from looking at the laws of physics in different reference frames that a given frame is in motion relative to any other all motion is therefore relative and also that different observers in different frames of reference while they’ll all agree that light moves at the same fixed speed regardless of their relative states of motion they will disagree on the lengths of objects and the durations of time that are passing in different frames these consequences will carry forward into the next section of the course a discussion of the Loren transformation and preview the conclusions that we’ll draw from the correct mathematics that relates observations from one frame of reference to another frame of [Music] reference in this lecture we will learn to appreciate the Gan transformation and it’s built in assumptions uh decent understanding of the past will help us to set the stage for the present we’ll learn a way to derive the form of the correct transformation between frames of reference respecting the postulates of special relativity and we’ll learn how to begin applying this transformation and see that it is in fact consistent with the postulates of special relativity it does end up being being entirely self-consistent and it gives us a basis for making predictions about the natural world predictions that can be tested the Galilean transformation was predicated on two assumptions and these assumptions may not have been made very clear when you were originally learned about this transformation for observers in inertial frames of reference that is frames of reference in which all observers agree that objects in motion are moving at constant velocities time is assumed to pass in the same way for all observers regardless of their state of motion and all observers agree that objects in each other’s frames are in states of constant motion and I’ve drawn down here an example graphic of a representation of an object in motion with some velocity Vector uh Illustrated here that I’ll use in a lot of the images going forward that will help us to think about these Transformations now let’s define two frames of reference that we can use as the archetypes for thinking about transformations of space and time information from one frame to another let’s denote one of these as frame s and we will always take frame s to be the thing that we call the rest frame now this is an arbitrary assignment you can choose one thing to be at rest and not another or vice versa but once you make that choice you need to stick with your choice you need to see that through to The Bitter End so for the purposes of illustrating the process of thinking about transformations of space and time information from frame to frame we’ll always take s to be the frame that is not in motion now in this frame of reference S we will imagine that they carry along with them a coordinate system uh like a framework of three lines that are at right angles to each other that they’ll use as reference markers for all spatial measurements and the coordinates from their cartisian coordinate system will be denoted X Y and Z when they describe object velocities they’ll notate them as using the letter U the letter V for velocity as you’ll see in a moment has a special place in relativity calculations and So to avoid confusing us as to what velocities we’re talking about we will use U to denote object velocities now we will Define a second frame s with a little Prime symbol next to it or S Prime that is moving relative to frame s at a velocity V so everything in that frame is moving all at once in the same direction at the same speed relative to S and in that frame they too have a little framework a little cartisian coordinate system framework of X Y and Z but they label their coordinates X Prime y Prime and Z Prime and when they measure object velocities they denote them as U Prime to be consistent with the notation of their coordinate system now we will do the following to simplify our thought process going forward it doesn’t have to be this way but we can set the problem up this way to make it easier for ourselves we will assume that they have arranged their coordinate axes so that they are always parallel to each other X is always parallel to X Prime even if x Prime is moving relative to X Y is paral parallel to Y Prime Z is parallel to Z Prime this makes it easier for us mathematically to relate things between the frames we can’t allow chaos to rign in all of this and we’re going to further simplify for the purposes of our discussion here that frame S Prime has a velocity V that is entirely and only along either X or X Prime it’s entirely parallel to X and xpre and has no component along y y Prime Z or Z Prime now in the Galilean picture of things in all frames time is absolute so that t is equal to T Prime the time measured in one frame frame s is equal to the time measured in another frame frame S Prime always that is the definition of absolute time now I’ve recovered the picture of our two little frames of reference here and our blue object in motion viewed from the perspective perhaps of one or the other frame with its velocity U or U Prime now this picture built from the postulates of the Galilean or Newtonian approach to space and time then allows us to define the equations transforming observations in one frame to observations in another frame for instance if we measure X Prime and T Prime and U Prime and frame S Prime these equations will allow us to figure out what the people in frame s would see and here are the equations you’ve probably seen them in this form or similar form in introductory physics because the motion of frame S Prime is entirely and only parallel to x and x Prime there’s a transformation between the X and X Prime coordinate system that is given by the first equation measurements in y are equal to measurements in y Prime and measurements in Z are equal to measurements in Z Prime and of course because of absolute time T is equal to T Prime now object velocities are related between the frames using the velocity transformation equation which if you use calculus and I would invite you to do this as a simple exercise you can prove from the first of the equations up here x = x Prime + VT you can prove that this equation is the addition of velocities derivable from the coordinate transforms using a little bit of calculus and a few minutes of of work on paper but basically the velocity observed for an object object in frame s is equal to its velocity in frame s prime plus v the velocity of frame S Prime with respect to S now there’s a problem here we know that the postulates of special relativity are more compatible with reality than the assumptions that were made to define the Galilean transformation in the first place and if you play around with a gilean transformation you can pretty quickly find out that it violates the postulates of special relativity so for example and this can be left as an exercise for the student especially because Maxwell’s equations are not something you get rigorous training on in introductory physics you can show that Maxwell’s equations their forms are not invariant under a Galilean transformation that would violate the first postulate of special relativity because if the equations if Maxwell’s equations have different forms that can be determined by experiment in different frames of reference that implies that it’s then possible to know whether or not you’re in the absolute rest frame for instance in the mechanical view of the universe in the frame of The Ether it’s a bit easier to see how the second postulate is violated by using a simple example you can imagine that the object in motion on the previous slides is a beam of light and it’s been emitted in The Ether frame which will take to be frame s the absolute rest frame and what you’ll find then is that that beam of light will have a very different speed in all other frames moving with respect to the ether frame the absolute rest frame or frame s in our notation here that violates the second postulate that the speed of light must be observed to be the same by all observers regardless of their relative states of motion so already the Galilean transformation is immediately shown to be at odds with the postulates of special relativity which again are based on observational evidence so here’s our picture again of these two inertial reference frames with a blue object being studied by both of The Observers in each of the reference frames S and S Prime and we want to find a transformation of X and X Prime Y and Y Prime v u and U Prime and T and T Prime between these frames um that gets something that’s compatible with the postulates of special relativity and we don’t have to change this picture to build up the correct transformation we just have to apply the postulat of special relativity in constructing the the transformation from one frame to the other and these new postulates the postulates of special relativity will enable us to arrive at a mathematics that’s consistent with observation so our goal is to figure out what is the correct transformation and we will continue to work with frames of reference wherein object velocities are observed to be constant that is inertial frames of reference that puts the special in special relativity now it must be true that in two inertial reference frames S and S Prime as depicted in the cartoon above that because the object in motion will be observed in either frame to have a constant velocity maybe a different magnitude but both frames of reference will agree yes we have each observed a constant velocity for the object that we’re studying it must therefore be true that X in the frame s is equal to the object velocity time time if it’s moving entirely along the X Direction and in the in the moving frame frame S Prime it must be true that X Prime is related to T Prime by The observed velocity of the object in that frame U Prime in order to further satisfy the first postulate of special relativity it must also be true that the transformation equations represent a linear transformation between the frames otherwise it can’t be true that all frames observe object velocities to be constant let me demonstrate this it’s important I think to start exercising your Calculus a little bit at this stage in the class so that you get a bit more comfortable with using calculus as a means to make predictions about the natural world so let’s begin by assuming and extremely generic form for the transformation between spatial coordinates in frame s and space in time coordinates in frame S Prime I’ve assumed that X is given by some unknown transformation with a spatial term and a temporal term each of these has coefficients I’ll talk about those more in a moment and each of the coordinates in the moving frame is raised to some unknown power for space it’s n and for time it’s m and similarly the time coordinate in frame s is related to the space and time coordinates in frame S Prime in a same way there are some new coefficients C and D that enter in here but again I’ve raised X Prime and T Prime to various Powers they could be two could be 10 could be 20 we don’t know now let me comment on these coefficients a b c and d are constants here we could always absorb some non-constant behavior in the coefficients into the function of X prime or t Prime that we’re using here now I’ve used a simple function just raising the uh space and time coordinates to a single power but you can draw the conclusion more generally that an arbitrary polinomial of X Prime and T Prime also won’t work to satisfy the requirement that all observers agree that the objects moving at a constant velocity regardless of their frame of reference in inertial reference frames so here’s my generic pair of transformation equations I don’t know what a b c and d are and I don’t know what n and M are but what I can do is I can recall that any velocity any object velocity like you know the object velocity along the X Direction in frame s uux or the object velocity along the uh X Prime axis in frame S Prime so U Prime X is defined by a derivative with respect to time that is ux is DxD t or U Prime X is DX Prime DT Prime that’s the definition of of velocity from its most foundational um aspects so what I would like to do to motivate that this has to be a linear transformation between the frames is to Simply take the above equations and turn them into statements about differentials of X and X Prime and T Prime and T rather than just statements about the coordinates themselves so this is where you can dust off your Calculus and see if you can arrive at the same answer that I get here but the bottom line is that the differential of X is related to differentials in X Prime and T Prime by this equation so the coefficients A and B remain unscathed but you wind up with this uh new power of X Prime and T Prime due to transforming this into a statement about differentials in space and time rather than just about space and time themselves and then similarly you get an equation that looks very much like that for DT differentials and time in the frame S as well okay so let’s hold those equations here for a moment and consider them so what I would like to do now is use these equations to relate The observed velocities in each frame of reference and so to do that I’m just going to take the ratio of the above two equations why because when I do that I get DX / by DT which on the left side is just the definition of the velocity of the object in frame s ux and on the right side I get something that’s a fair bit nastier than that but we will simplify it into something that looks a bit more familiar in a moment so notice that I have DX Prime and DT Prime both in the numerator and in the denominator of this ratio so what I can do next is I can divide the top and the bottom of the right hand side by DT Prime doing so eliminates DT Prime from the right hand terms in each part of the ratio and creates a DX Prime DT Prime in the left term of each part of the ratio and that should look very familiar because DX Prime DT Prime is by definition the velocity of the object in the prime frame in the S Prime frame and so we arrive at this final relationship that relates the observed velocity of the object in frame S Prime to The observed velocity in frame s but there’s a problem here unless Nal m = 1 the above equation will always leave a lingering space and time functional dependence on the right hand side which violates the first poti of special relativity the speed observed in frame s even if the speed in frame S Prime is constant will not be observed to be constant because it will depend on where in frame S Prime the object is at any given moment it will have a space and time time dependence that that is uh rather nonlinear and so we’re forced to conclude that in order to be compatible with the basic idea that we’re looking at inertial reference frames and relating object velocities uh in inertial frames of reference where there are no net forces that can cause changes in in the state of motion of the object we’re really forced to choose a linear transformation from frame to frame now that’s what we had in the Galilean transformation but we are still stuck with it even here in special relativity that’s a good thing because vastly simplifies the mathematics okay so we’re going to now begin to build up the mathematics of the transformation now that we’ve accepted that we need a linear transformation from frame S Prime to frame s for example we get a very simplified pair of equations x = a x Prime + BT Prime and T = CX Prime + DT Prime but we don’t know what these coefficients are they may be trivial they may be zeros or ones but we need to figure it out so now that we’ve established the linearity of the transformation we need to nail down a b c and d so we need to think of some special limiting cases of this picture where we can isolate these constants maybe one at a time or in small batches and in doing so figure out what they are in order to be compatible with the postulates of special relativity now this is a standard trick in algebra we have four unknowns in two equations we’re going to need four special cases to solve for all the unknowns and the postulates give us the framework to define those special cases so let’s pick special case number one where we take the moving object the blue ball and we pin it to the origin of the coordinate system of frame S Prime so in frame S Prime the object will always be located at 00 0 in the X Prime y Prime Z Prime coordinate system it’s moving along at the same same speed as the frame itself and so it’s observed to be at rest in frame S Prime in frame s however what we see is we see the blue ball pinned to the coordinate system of this Frame and they’re all moving together at a velocity V to the right along the x axis so in frame s it’s observed that the object is moving at V so U equals V in frame s now when we do this we have a simplifying situation for X Prime X Prime will always be zero because this thing is pinned to the origin of frame S Prime and so we can simplify the above equations to the following we have now that X is just equal to BT Prime in this special case T is just equal to DT Prime in this special case and then the velocities x equal U and 0 equal U Prime T Prime are the resulting equations from this special case so let’s take the first two equations the one for x and the one for T and divide them and then let’s use the third equation xal UT as a substitution to eliminate one of the unknowns so when we do this we wind up with x / T is equal to B over D go ahead and check this yourself and then from the velocity equation we get that x/ t is equal to U but in this special case the object speed is also the frame speed V so we wind up with x/ tal V and as a consequence of that we get the first constraint on our coefficients whatever B and D are their ratio is equal to V the velocity of frame S Prime now let’s choose a second special case and you might have guessed that this would be the next thing that we would do we we pin the blue object to the origin of the rest frame coordinate system we put it at 0000 0 in frame s so x y and z are 0 0 and zero and now we observe that blue object from frame S Prime now from the perspective of frame S Prime which is moving to the right at speed V the blue object appears to be falling further and further behind their frame its velocity appears to be negative V from the perspective of an observer in frame S Prime the moving frame so with that in mind and fixing X to Zero in the uh General equations up here on the right we can simplify the equation set to the following 0 = ax Prime + BT Prime uh the T equation doesn’t get affected by any of these choices we wind up with 0al u and X Prime = VT Prime substitut suting in for U Prime with Nega V now let’s employ the first and third equations namely this simplified first equation for the coordinate X and this x Prime equal Nega VT equation to further get the constraint on coefficients so if we do that we wind up with the first equation telling us that negative ax Prime equal BT Prime that’s the consequence of the first equation from the third equation we get that X Prime is negative VT T Prime and if we combine these two things together we find out that avt Prime equals BT Prime now T Prime entirely drops out of both sides using the substitution and we find out that V equal B over a go ahead and try this yourself I’m going through this a little bit fast but of course you can pause this at any time and work through the algebra on your own and so we arrive at our next batch of constraints now we already knew from the first special case that V equals b/ D from the second special case we also find out that V equal b/ a and if these two constraints are simultaneously true then it must be true that a equals D so now we have really constrained ourselves down so here’s the third case we’ll look at what if the object is a beam of light now this is the first time that we will definitively deploy one of the postulates of special relativity specifically the second postulate because if the object that’s observed to be moving in both frames of reference is a beam of light then by the second postulate of special relativity observers in both frames must observe the velocity of the object to be exactly C 2.9 98 * 10 8 m/s regardless of their relative motion so what happens to our equations as a result of this fact that according to observations and encoded in the second postulate all observers observe that the beam of light moves at the same speed regardless of their state of relative motion the equations simplify as follows the first two don’t really change at all but because the last two are statements about velocity and their relationship to space and time measurements it must be true that x equals CT but also X Prime equals c t Prime when the object in motion is a beam of light so let’s combine the first two equations the x and t equations and substitute using the uh information from the third equation at the bottom and when we do that this one gets a little nasty at first we wind up with x / T equals this horrible ratio over here not looking very promising so far but X overt is just C by the third equation and if we divide the top and bottom of the ratio on the right hand side by T Prime we wind up with X Prime over t Prime in both the numerator and the denominator and X Prime / T Prime is just C the speed of light so we wind up with the speed of light here the speed of light here and the speed of light Here and Now what we can further do is take the previous constraints relating b and d and a we can substitute those in and go through a simplification process and when we do that we find out that c is equal to B over the speed of light squared and we finally arrive at C = A * the velocity of frame S Prime relative to S over the speed of light squared so all of our constraints allow us to eliminate B eliminate C and eliminate D from the equations up here in the top right of the screen B is equal to AV C is equal to a * V over the speed of light squared and D is just equal to a itself so what we have done now with special case 1 2 and 3 is we’ve eliminated three of the unknown coefficients in favor of the fourth a and all we have to do is come up with one more constraint that allows us to figure out what a is well here’s our last case and the last case is a basic assumption about the Transformations first of all we chose that the transformation of x and t to X Prime and T Prime from the perspective of observation in frame S Prime being mapped onto observations in frame S have a certain form but because it shouldn’t matter which frame we pick to be the one that’s at rest and the one that’s in motion we should get the same transformation equations if we had started with frame S Prime being at rest and having frame s be the one that was in motion and the only thing that should change between observations in frame s and observations in frame S Prime is that the Rel relative velocity of the two frames changes sign that’s the only thing that should change when you alter the perspective of which one is at rest and which one is in motion and so as a consequence of that we should be able to eliminate the unknown a and figure out what it actually is so let’s start by writing down x and t in terms of the coefficient a and all the other things we’ve already sorted out so I’ve effectively just copied these two equations down here next let’s rearrange and rewrite these equations not as solutions for x and t but solutions for X Prime and T Prime as if we were trying to figure out what the person in frame S Prime would have seen if frame s was chosen to be the frame that was in motion now a lot of algebra is involved in this and I will leave it as an exercise to the viewer to try this out to practice their chops at algebraic manipulation in order to get what we want but the bottom line is that if you work this through you will find out that X Prime is given by this nasty function of x and t and t Prime is given by this equally nasty function of x and t now these equations tell us what should have been observed in frame S Prime given observations in frame s treating frame S as if it had been the frame in motion and frame S Prime as if it had been the frame at rest but all that should change when switching perspectives on the problem like this is you should get the same equation differing only by a minus sign on any term with v in it then we’re forced to say that a is equal to 1 over a * the quantity 1 – v^2 over c^2 theus1 and if you rearrange now and solve for a you find out that a must be equal to 1/ the < TK of 1 – V ^2 / c^2 this quantity shown here now this may not look pretty but this strange thing 1 over the < TK of 1 – v^2 over c^2 shows up all over the place in special relativity calculations and so it’s given a special name we don’t leave this as coefficient a it gets the symbol gamma the lowercase Greek letter gamma shown here in the lower right gamma is defined to be this strange Beast here one over the < TK of 1 – v^2 c^2 so let’s take a look at the final form of the Loren transformation the mathematical transformation that obeys all the postulates of special relativity if we are making observations in frame S Prime and relating them to frame s which is taken to be at rest then these equations here tell us what we want to know we measure things in frame S Prime and we get X Prime and T Prime Prime and then we combine them using these equations to get x and t the observations that should be made in frame s if on the other hand we’re making observations in frame s and want to convert them to observations in frame S Prime we flip the sign of v and change all the coordinates around and we get basically equations that look the same up to a minus sign on terms that just have V in front of them this number gamma always lurks out in front of everything and we see here another interesting thing that’s going on we see that in transforming space and time from one frame to another the space and time measurements in one frame get tangled up to compose the space and time measurements in the other frame in a sense space and time and special relativity are not separate entities we we treat them as separate in introductory physics but the lesson of special relativity is that we should have been thinking of this all as one framework SpaceTime not separate Frameworks of space and time this entire time now this multiplicative Factor gamma depends on the relative velocity of the two frames and as you’ll see it’s effectively a measure of the degree of the relativistic effects how much you need to take into account special relativity to solve a problem correctly between two frames of reference and again it’s given by this funny combination of velocity of the frames relative to each other and the speed of light one over the square root of the quantity 1 – v^2 over c^2 so let’s build a little bit of intuition about the meaning of the gamma Factor it appears everywhere in relativity calculations at least in special relativity it gets absorbed into other things and what is known as general relativity the general theory of space and time which we will only cover in the most shallow way in this course it’s largely un avoidable for all of the physics calculations you’re going to do going forward so let’s try to understand it a little bit better and to do that I I think we can build some intuition by playing around with the quantity gamma at various limits of its uh observable nature so for instance what is gamma for a frame S Prime that’s at rest with respect to frame s well we would expect to find that the two frames are the same since they are then in the same state of motion well we already know that if frame S Prime has a velocity of zero with respect to frame s that the terms with VT in front of them will vanish but what happens to gamma well if you plug in zero for V in the function gamma you find out that indeed for V equals z we observe that gamma is one in other words the multiplicative factor in front of either the space or the time coordinates when relating those to space and time coordinates in frame s all become coefficient of one in other words you’re in the same frame so you should get the same space and time measurements that’s good that’s what we would hope would happen now on the other hand what is gamma for a frame S Prime that achieves a velocity of exactly C the speed of light relative to S so this would be like imagining a frame of reference that’s pinned to a beam of light moving at the speed of light and it’s another very weird special case and it’s weird because what happens is that the gamma function takes on its biggest possible value Infinity you wind up with 1 over the square root of the quantity 1 – c^2 over c^ 2 well c^ 2 over c^2 is just 1 so you wind up with 1 over the < TK of 1 – 1 which is 1 over Z which is infinity zero goes into one an infinite number of times so that’s the upper limit for gamma so far as we know it’s impossible to travel faster than the speed of light there’s no observational evidence that anything does travel faster than the speed of light and so we are led to believe and special relativity encodes this that the fastest speed in the universe is that of light and so gamma takes the special value of infinity at that speed so as we can see gamma is a frame velocity dependent number and it has a well-defined range at the low end its smallest value it can take is one and at the high end the largest value it can take is infinity and it can take all numbers in between that depending on what V is I think it’s useful to graph this albeit perhaps in a way that’s not terribly familiar to you this is a plot of the value of gamma so the so-called gamma Factor on the y- AIS as a function of the frame relative velocity V on the x axis and so you can see here that I have chopped off the low end of the xaxis at about 10% the speed of light why because gamma has a value that’s so close to one that generally speaking you don’t have to worry about it being different from one now that’s not true in all cases and we’ll look at some of those cases going forward in the class but generally speaking if you are at about 10% or less the speed of light you do not expect to really observe what are called relativistic effects that is effects that distinguish an observation from what you expected from Galilean or Newtonian relativity above 10% of the speed of light however gamma can begin to take values that are appreciably distinct from one and you can see here that when you get to values that are about half the speed of light which occurs roughly here on this graph this is 1 * 10 8 this is 2 * 10 8 so this is about 1.5 x 10 E8 right here and as a result of that you can see we’ve now appreci appreciably started getting gamma factors that are above one by about 20% or so when we get to about 2/3 the speed of of Light 2 * 10 8 m/s we’ve achieved gamma factors that are about 50% bigger than 1 so 1.5 and as we begin to approach closer and closer and closer to the speed of light we see that the gamma function takes on increasingly larger and larger and larger and larger values spiking up to Infinity at exactly the speed of light this plot will help you to understand why it is that we just didn’t notice the these deviations from the Galilean or Newtonian view of the universe for most of the history of Science in humanity and that is because the laboratory experiments that we were effectively conducting as a species were all done at speeds that were far less than that of light and so we really never would have noticed these effects to begin with it was only when we began playing around with light and things that can

move very close to the speed of light like subatomic particles that we began to get ourselves into trouble with the intuition we had built up on Human Experience prior to that but we see now that the postulates of special relativity predict that we should have expected deviations from that Newtonian or Galilean view of time where time is the constant between all observers it’s not it’s the speed of light we can see this effect encoded in the gamma function now as has been teased in the previous lecture on the basics of special relativity that is the postulates and their con consquences this theory of space and time has some consequences that can feel surprising to the average human being for instance objects in motion relative to what we consider to be a rest frame that is the frame we denote s will appear contracted along the direction of travel we can actually show this as a prediction of the Len’s transformation now that said to appreciate this particular effect even from the Loren transformation really requires you to think extremely carefully about what it has ever meant to measure the length of something I feel that discussion is best saved for class time as in class time we can get very handson with the concept before we start plunging into calculations where the language you would use to describe the recipe for attacking this particular question may not feel very natural because you haven’t really thought about what it means to measure the length of something especially the length of something that’s moving relative to you so instead in this lecture let me concentrate on frames in motion relative to the rest frame s which will also observe a passage of time that relative to S seems slowed this effect I labeled time dilation and we’re going to formally calculate it now and finally I’ll also look look at events that are simultaneous in one frame and show that that simultaneity is not guaranteed at all in another frame that’s moving relative to the first let’s explore time and simultaneity in this lecture so how does one consider the passage of time in different frames of reference well to measure time we have to define a clock of some kind a regular pattern of events that all happen for instance at the same reference point in in a frame so consider a clock that’s at rest in frame S Prime and it provides regular information so for instance pulses of light at different times T1 Prime T2 Prime T3 Prime and so forth always with regular intervals between them but that clock is always pulsing at the same position X Prime so X1 Prime equals X2 Prime equals X3 Prime whenever the time measurements are established and what we want to know is what’s the time between the pulses observed in the rest frame so that clock is in a frame that’s moving with respect to a frame that we agree is at rest we call it to be the rest frame what does a observer in the rest frame observe the time to do in the moving frame well again you want to relate time observations between the two frames but to do that you need to use the Loren transformation which takes space and time information from the moving frame and translates it into time information in the rest frame so we want to take the the pulse at T2 and transform its time into what the uh person in the rest frame measures they’re so-called T2 and we want to take the Pulse at T1 and transform that into the rest frames T1 and to do that we have to use this equation this comes from the Loren transformation now we have a simplifying fact here and that is that the clock in frame S Prime is always pulsing away at the same location X1 Prime equals X2 Prime so if we were to combine these two equations to calculate the duration of time between T2 and T1 we might do the following we might take T2 minus T1 and try to figure out what that is in terms of T2 Prime minus T1 Prime well because the clock pulses at the same location in frame S Prime X2 Prime and X1 Prime terms cancel out and we’re left with this equation which relates the durations and time observed in the two frames by a gamma factor and so I can write the two time durations delta T in frame s the rest frame and delta T Prime the the time duration and the moving frame and I can relate them and they’re related Again by a gamma factor and I find that if I take the ratio of the time duration observed in the rest frame and the time duration observed in the moving frame that they will differ they will not be a ratio of one and the ratio however will be given by the gamma factor which takes a value of one but only in the the special case that the two frames are at rest with respect to each other at any relative speed greater than that gamma takes a value that’s greater than one now until you get to very high speeds it’s not appreciably greater than one but nonetheless it’s not exactly one unless you’re at rest with respect to each other and so that we see now that in a frame that’s moving relative to another durations of time will always observe to be shorter than in the rest frame the duration of time observed in the rest frame is greater than the duration of time that’s observed in the moving frame for the same pair of events and the degree of dilation of time depends again on the ratio of V over C specifically through the gamma Factor time in the moving frame will appear to the rest frame to pass more slowly now another expected effect due to special relativity is that events that are simultaneous in one frame of reference may not necessarily be simultaneous in another now we already explored that a little bit even under classical velocity situations but we can revisit that idea here under the Loren transformation so for instance consider two events like pulses of Light which are observed to be simultaneous in frame S Prime the moving frame the events have coordinates X1 Prime T Prime and X2 Prime T Prime so what is the time between the events observed in the rest frame does the rest frame observe that they are also simultaneous that is also at exactly the same time well we can start by relating the times in the rest frame to the space and time measurements in the moving frame for T2 and T1 again we’re just writing down the Loren transformation here between observations in frame S Prime and the observations we want to establish in frame s now since the events are simultaneous in frame S Prime T1 Prime will be equal to T2 Prime so if I now calculate the duration of time that passes in frame s T2 minus T1 I find a very interesting fact that it’s not equal to zero it’s equal to this thing on the right hand side here which depends on the velocity of the frames relative to one another V quite directly in this case not just through the gamma Factor but gamma multiplied by V and what’s particularly interesting about this is this question whether two events are simultaneous in one frame and simultaneous in another frame of reference that’s in motion relative to it really picks at this interesting thing I mentioned earlier which is that space and time have to be treated as one framework space time and they can get tangled up in each other and what we see here is that because the events in frame S Prime are simultaneous but not necessarily at the same location in space in frame S Prime this creates a displac m in time between the two events in the rest frame in other words delta T in the rest frame is not necessarily equal to zero it’s equal to gamma V / c^2 times the spatial displacement of the two events in the moving frame Delta X Prime so we see that in the rest frame events cannot be observed to be simultaneous even when they are simultaneous in the moving frame unless those events happen at exactly the same position in space that is X2 Prime equal X1 Prime that’s a special case or unless the two frames are not in relative motion to each other that is V equals z in that case of course the two frames become indistinguishable and the whole discussion was moot to begin with but if the two frames are not the same frame of reference if the events occur at different locations in space in one frame but are otherwise simultaneous there they will not be viewed to be simultaneous in the other frame simultaneity of events could have been guaranteed in Galilean relativity even if they were at different um locations in space because of the absolute passage of time but since time is not absolute and special relativity doesn’t accept that as one of its postulates you find out that except under these very special conditions simultaneity cannot be guaranteed in another frame and again this is a really good example of how space and time are not Inseparable from one another in special relativity in trans forming observations in space in one frame you wind up with observations in time in another frame space and time get kind of Tangled Up in each other and going from frame to frame a spatial separation in S Prime becomes a temporal separation in frame s and I find this to be one of the more remarkable features of space and time as viewed through the lens of special relativity now one last question we can visit in all of this is whether or not it’s possible to recover classical physics from this view of the universe in other words are the Galilean Newton view of space and time and relative motion totally gone were they totally wrong this whole time it turns out the answer is not really after all the Galilean transformation did work in real computation for centuries before special relativity was needed right I mean people were able to relate observations in different frames of reference at relatively modest speed compared to those of light so one of the things that that should become evident from all of this and this is a general feature of a good theory of nature a good predictive description of nature that can be tested and even falsified um a good theory of nature describes all new phenomena but also it accounts for the existing confirmed observations the old observations the old phenomena that we had all that experience with from which we built intuition what generally seems to happen is that when you find out that your current understanding of nature is wrong you find out eventually through enough observation and experiment and mathematical work the correct description of Nature and you find out that your old observations were correct but in a more limited regime of nature in this case for instance low velocities so to recover the Galilean or Newtonian view we need only slow nature down from speeds close to that of light for example we can consider the special case of speeds between the frames of reference that are much much much much much much much less than the speed of light so that V over C for instance becomes a very tiny number approaching zero so let’s look at what happens to the gamma factor using something called the binomial expansion now I’m going to illustrate the binomial expansion for this specific function here but in general there is a general form for the binomial expansion and you can use a math reference on the web or a paper book to explore the binomial expansion more on your own free time so let’s begin by looking at the gamma function the gamma function was defined as 1 over theare < TK of 1 minus the quantity V ^2 over c^2 so what we have here is we have a function of a number v^2 over c^2 that’s bounded between 0o and 1 V over C can either be Zer for V equals 0 or 1 for V equal C and as a result of that gamma takes values between one and Infinity but this this parameter V over C that depends on the relative frame speed that thing is bounded between zero and one now one of the things that we can do is we can rewrite this gamma function as a more basic generic looking function by replacing V over C with a number alpha or in other words rewriting this in terms of alpha s which is v^2 c^2 and note that Alpha squ is a number that is less than or equal to 1 and its lowest value it can take is zero so we wind up with this more generic looking function 1 – Alpha 2 all raised to the- one2 power in other words 1 over the < TK of 1 – Alpha 2 now if we apply the binomial expansion to this more generic looking function with this condition on Alpha that it’s bounded between zero and one then we find that we can rewrite this function as a series expansion the first term in the series is just the number one the second term in the series depends on Alpha and it’s plus 1 12 Alpha squar now the binomial expansion allows you to keep adding terms that have higher and higher powers of alpha in them the next one will have a power of alpha to the 4th the one after that Alpha to the sixth and so forth with different coefficients in front of them now because Alpha is a number that cannot be greater than one as a result of that for the special case that Alpha is much much much less than one in other words as Alpha tends towards zero we see that indeed we recover the behavior that as these terms with alpha squar alpha to 4th Alpha to the 6th as they approach zero the only term left that really dominates in the sum is the leading term one and we see that gamma is approximately equal to one as Alpha gets closer and closer and closer to zero and when Alpha equals Zer we get gamma equals 1 which we know is the limiting case of gamma for velocities of zero speed so that makesense it’s approaching the limit in case when V equals z that’s what it means for Alpha to go to zero it means V is going to zero too so what happens to the luren transformation equations when we replace gamma in it with this binomially expanded version of gamma so let’s start with xal gamma * the quantity X Prime + VT Prime let’s substitute in the binomial expansion in terms of V over C being equal to Alpha all right so now we replace gamma with this thing 1 + 12 Alpha squ plus all terms with Alpha to the 4th and higher in them and in the limiting case that V is much much less than C that is as Alpha approaches zero these terms with alpha squar alpha to 4th Alpha to the 6 they contribute less and less and less and less to the sum until we’re left with just one in front of the sum X prime plus VT Prime so in the special case that the velocity we’re considering for the relative motion of the frames is much less than that of the speed of light we find that we recover x = x Prime + VT Prime which is the old Galilean relationship between x and x Prime and T Prime now similarly I can take the equation relating X Prime and T Prime to time in the rest frame and I can substitute for the gamma factor using this binomial expansion I can also notice that there is a v over C lurking here in front of the X Prime coordinate that’s another Alpha that sits in front of X Prime so if I write that all out here’s the binomial expansion of gamma here’s that Alpha that I’ve substituted in for the V over C that was lurking in front of X Prime and what you’ll notice is if I distribute this gamma to both the terms inside of this sum that the space term the X Prime term always has an alpha somewhere in front of it multiplying it you can’t escape it you don’t just get a bare number like one multiplying the X Prime coordinate whereas for the T Prime coordinate there is a term in the expansion that just goes as 1 * T Prime and so in the limit that the velocity is much much less than the speed of light all terms with Alpha in front of them vanish to zero and we’re left just with t Prime in other words in the low velocity limit T is equal to T Prime and we see that we have completely recovered the Galilean transformation and we’ve reconciled with classical physics in the limit of low velocities this is why time appeared to be absolute in the original formulation of mechanics it’s because when the velocities are much lower than the speed of light between two frames of reference you have a very hard time seeing these extremely subtle effects between clock measurements between the two frames but that is laid bare as a false perception of nature as you approach the speed of light in relative velocities between two frames of reference but we see that we can Recon the old picture of space and time with this modern and correct picture at least correct as regards observation of the natural world simply by considering the limit of of small velocities compared to that of the speed of light and we completely recover the old view the old view is nested inside the modern view as a limiting case so to review in this lecture we have learned the following things we’ve learned to appreciate the Galilean transformation and the assumptions upon which it’s built we’ve learned a way to derive the form of the correct transformation between frames of reference the so-called Loren transformation that is the modern way of relating observations in one frame to observations in another frame of reference and we’ve begun to see how you start to apply this transformation by asking questions like what are the events and in what frame are they defined and is anything the same for those events are space measurements the same are time measurements the same in a given frame of reference to simplify the questions that we’re trying to answer and then get the answers out of the luren transformation and we see that we have arrived not only at a transformation that’s consistent with the postulates of special relativity but which gives us a mathematical formulation for the intuition that we built off of the postulates that distance and time measurements are not going to be the same in different frames of reference even if all observers agree on on the speed of light as a constant of nature so while we see that all observers must agree that light moves at the same speed regardless of their relative motion nonetheless observers in different frames of reference will disagree on lengths of objects the durations of time that pass and the simultaneity of events events simultaneous in one frame will not necessarily be simultaneous in another frame of reference except under very particular conditions we’ve also seen how to recover classical physics from special Rel ity by allowing the velocity uh of the relative motion of the two frames of reference to drop far below the speed of light so that these Corrections from the the original Galilean transformation all vanish and leave behind the Galilean transformation with its Assumption of absolute time laid bare and we’ve seen how in that limit the Loren transformation exactly reproduces what were the original assumed relationships between space and time and velocities as encoded in the gy transformation we’ve recovered the past from the present and we can continue to use the present to build a foundation for making future predictions and that is precisely what we’re going to do in the next section of the [Music] course in this lecture we will learn the following things we will learn what is a muon we will learn how to use the muon as a laboratory for making predictions with the Loren transformation and finally we will learn how the muon was the first direct test of the validity of special relativity let’s begin by reminding ourselves one more time time about the Loren transformation if we make observations in a frame S Prime that we consider to be moving and we want to convert them into observations in a frame s we consider to be at rest then the equations in the top left of the slide will do just fine if on the other hand we have observations that are made in the rest frame s and we want to convert them into observations in the moving frame then all we have to do is change X to X Prime T to T Prime and V to negative V in the upper left equations and we get the necessary equations in the upper right this function gamma that appears in all of these equations is a function that depends on the relative speed between the two frames V and it has the form of 1 over the < TK of 1 – V ^2 over c^2 where C is the speed of light this can be expanded into a series representation of this function using the binomial expansion we’ve looked at this before and so we get an expansion that looks like 1 plus a half v^2 over c^2 plus terms that are higher order in V like V 4th V to the 6 and so forth and for sufficiently low velocities less than a few perent the speed of light we can usually neglect some or all of these higher order terms in the series expansion and get a simpler representation of the gamma function at low velocity now we’ve looked at some consequences of the Loren transformation and special relativity we’ve looked at length measurements and we’ve seen how they depend on frame of reference we’ve looked at time measurements and seen how they depend on the frame of reference and we’ve looked at the simultaneity of events in one frame and see that they are not necessarily guaranteed to be simultaneous in any other frame that’s moving with respect to the one in which they are simultaneous but is the relativity of time a real thing I ideally what we would want to do is take two twins put one in a spaceship that can accelerate very quickly up to speeds well in excess of half the speed of light that’s where the gamma function begins to take on values that are very much in excess of one and then we would let the twin travel out on a journey of maybe 10 or 20 years and then bring them back that’s going to take another 10 or 20 years and when the twin gets back from the journey at high speed we would compare the two twins and see how they have aged from the perspective of observers on Earth that would be a great experiment except that it is really not easy to construct a vessel not only they can hold humans for long duration space flights but also that can accelerate up to speeds that are appreciably close to that of light this is an engineering challenge that we as human beings have never really mastered instead what we need to do to test the claims of special relativity is to identify a laboratory where such speeds can be readily achieved but also one where there’s a natural clock of some kind a regular sequence of events whose time can be well predicted so that we can compare those things when they’re at rest to when they are in motion now tiny particles would be a great potential lab atory tiny particles have very small masses they’re very easy to accelerate and so it’s possible that something like the electron with a mass of 9.11 * 10- 31 kg could be ideal for investigating fast moving objects and maybe even the relativity of time you know for example the ground state energy of the hydrogen atom the lowest energy state that an electron can have when orbiting a single proton is 13.6 electron volts you can look back at the conversion factor for the electron volt look it up on the web whatever you want to do but you’ll find that this comes out to be about 2.19 * 10- 18 jewles of energy now if you translate that into a corresponding kinetic energy for the electron in that state of hydrogen what you’ll find is that you can calculate the speed of the electron that is represented if you pretend that the electron is orbiting like a little planet around the proton under the influence of the kulum force so let’s do that let’s imagine that the electron is orbiting the proton at the center of the hydrogen atom and let’s use the energy of this state to estimate the kinetic energy and therefore the velocity of this electron and if we do this using the classical kinetic energy 1 12 MV squ and rearranging it to solve for the speed of the electron we find that it should should come out to be about 2.2 * 10 6 m/s or thereabouts that’s already pretty fast on its own that’s about 1% of the speed of light without having to do anything exotic except maybe study the electron in a hydrogen atom now of course we’d like to get the electron up to faster speeds than that but if but if that’s how fast it’s already going in a hydrogen atom then you can imagine it’s probably not too hard to get it going faster in fact JJ Thompson who’s credited with the discovery of the elron um isolated them by ripping them off of parent metal atoms using the kulon force using a strong electric field and a large electric potential this has the effect of accelerating the electrons up to rather High kinetic energies for the tabletop experiments of his day representing tens of thousands of electron volts of energy in the electrons and that would equate to speeds roughly of the scale of 108 m/s perfect those are the speeds we want to investigate phenomena at so a particle like the electron would be easy to accelerate but there’s a problem the electron doesn’t do anything it’s an extremely stable particle in fact left on its own an electron will simply be for the rest of the history of the universe so far as we know so it doesn’t have any regular characterizable phenomena associated with it once you’ve isolated an electron it lacks a kind of clock that it carries along with it that we could use to see whether or not the passage of time is affected by the motion of the electron well are there any such clock-like phenomena in nature that are associated with very small particles the answer is yes radioactive decay of atomic nuclei is exactly an example of a natural clock that ticks all the time in nature whether we’re there to observe it or not and if we do observe it we can use it to measure the passage of time in a system so for example among her many discoveries two-time Nobel Prize winner Marie curri isolated the element polonium it is highly unstable and the natural isotope of polonium polonium 210 transforms spontaneously into a stable lead atom lead 206 after emitting energy in the form of radiation specifically what it does is it spontaneously ejects two protons and two neutrons from the polonium nucleus these two protons and two neutrons are bonded into the nucleus of a helium atom and this thing is known as an alpha particle we’ll return to alpha particles later in the course the bottom line is there’s some spontaneous phenomenon that happens with regular time intervals that we can use to actually track time in nature now polonium 210 has what is known as a halflife of 138 days but what does that phrase halflife mean it means that if I were to isolate 100 atoms of polonium 210 in a sealed container and have some way of looking at those atoms and Counting them every hour of every day if I were to wait 138 days from the time I seal the container and then look in the container on average I will find that after 138 days about 50 atoms of polonium 210 will remain in the container the container will also now be home to 50 lead 206 atoms they resulted from the spontaneous decay of the missing polonium 210 now if I further wait another 138 days from that moment and look in the container again on average I will find that I now have 25 polonium 210 atoms left half of the sample I had 138 days ago and correspondingly I’ll find 75 lead 206 atoms in the container a halflife is a regular interval of time and if you had some kind of equipment that could be used to establish the amount of a certain isotope present in a sample you will find that after one halflife after every halflife passes you’ll lose half of what was there the last time you looked so unstable radioactive elements have a reliable built-in clock a regular process that occurs at the same place that is the nucleus at regular time intervals however there’s a problem and in the historical context what I’m talking about here is is a problem in the early 1900s polonium and other radioactive elements were pretty hard to come by in the days when they were discovered and even in the the decades after that and even if you could isolate an appreciable sample of them how would you know precisely how to count the numbers of those things whether they’re at rest or whether they’re in motion and and not only that you got to put them in motion which means you need to accelerate them and they are thousands or tens of thousands or hundreds of thousands of times heavier than electrons particle accelerators that are capable of bringing ions up to speeds approaching that of light of any decent quality and control were decades away in the early 1900s they wouldn’t emerge until the 1930s 40s and 50s so it’s nice that we have these regular clocks like radioactive isotopes but you can actually do practical experiments of the variety we’re thinking about trying to do that is attempting to see whether their clocks slow as they are put into a state of motion relative to the observing frame if only we had a tiny particle that combined the lightness abundance and ease of acceleration of the electron with the regular instability of radioactive atoms and it’s into this part of our story enters the muon or for short the MU now the mathematical description of unstable nuclear behavior and of the strong binding of things in the nucleus took decades to work out but around the 1930s with some experience now with other forces in nature like electromagnetism it was hypothesized that the forces inside the nucleus that both bind it together so tightly but also occasionally allow it to catastrophically break apart that these forces were maybe of two different kinds and that they had particles like the particles of light that transmit electromagnetism that acted as intermediaries in the nucleus and transmitted these forces within the nucleus and so these intermediaries were given a generic class of name maisons from the Greek word MOS meaning intermediary and and by the 1930s or certainly the 1940s the hunt was on to find them now shown at the left here on this slide is an image that was taken by two physicists Anderson and netm and published in 1936 a previously unobserved electrically charged particle punches through the slab of lead that runs through the center of the photograph these are two different views taken from different angles of the same particle interactions at the same moment in time and the interactions are taking place in a lead Target that runs through the center of the picture roughly here in the picture now as this previously unidentified particle passes through the lead it knocks apart nuclei but in this process it barely loses any energy this was a really strange beast in its day it would come to be dubbed the muon or muon for short as the physicists of that day mistakenly thought that this has to be one of the sought-after nuclear force intermediaries I mean what else could it be this turned out to be a bit of a lack of imagination and experience on the part of physicists with the broader picture of nature a good lesson for all of us of course and this assumption turned out to be wrong the particle was real but its role in nature was not as originally assigned and that wouldn’t be fully understood until the 1940s and 1950s its electric charge however was pretty well determined from careful experimentation and it was found to be identical to that of the electron ne1 . 609 * 10-9 kums so it carries with it the same Elementary charge that the electron possesses its mass however was very unlike the electron it weighed in at 27 times the mass of the electron too light to be a proton too heavy to be an electron and crucially unlike the proton and unlike the electron it is also so unstable if you trap a muon nearly at rest and there are some fairly straightforward ways to do this on average you will find that it only lives about 2.2 microsc or 2.2 * 10- 6 seconds now let me make an important aside while we’re on the topic of unstable particles about unstable particles half- lives and the characteristic LIF time of an unstable particle the mathematics of unstable particles was developed in response to the discovery of radioactive decay of atomic nuclei and it’s it’s a fairly straightforward application of algebra and calculus and I find it’s instructive to run through it here consider the kinds of systems we’ve been talking about so far so for instance let’s imagine you have a system of n0 unstable objects you know like a 100 nuclei or particles like the muon and You observe them to have that number at time T and then you wait a little bit we consider some change in time t plus delta T at which point we then discover that the number of objects has decreased by negative DN now here DT and DN are differential units of time and number respectively and you find that the number of objects remaining after a time has passed DT since the original time measurement is the original number n0 minus DN but if you double the number of objects objects so if you start with 200 unstable objects for instance and wait the same amount of time you don’t find that the DN is the same size as it was before it gets bigger and if you triple the objects to 300 or quadruple them to 400 you again find that DN after the same DT is proportionally larger and it’s larger in proportion to the size of the starting sample there is some proportionality between the change of number of objects the change in time and the original number of objects and so we can express this observational relationship in a simple equation negative DN the change or decrease in the number of objects is equal to a constant which we have yet to determine we’ll denote that the with the Greek letter Lambda lowercase Lambda times the original number of objects n times DT the time that passed during which the time the number changed now you’ll notice that this is set up to look like a an equation of differentials and so one could actually integrate both sides of this equation you can put all the n’s and DNS on one side and all the constants in DT on the other side and then you could integrate the side with the number stuff on it from the original number n Zer to the final number n after waiting a Time T now you’ll notice that on the left hand side we have the integral of -1 Over N DN or 1 /x DX if that sounds more familiar to you and so you you should know from some experience with second semester calculus that the natural log of the argument n in this case will wind up being the answer to this integral on the right hand side we have a much simpler integral we’re just integrating a constant times DT from time Z to time T whenever we observe the system later and that’s a very simple integral you just wind up getting the time T back times Lambda and then you just evaluated at the end points so if you do that you should find that you get the following equation the natural log of the original number of objects minus the natural log of the final number of objects at time T is equal to Lambda * T minus 0 zero is the original time at which You observe the system and see that it has objects n zero in it well if you rearrange now and try to isolate the number of objects n at the later time t on the left side you wind up with this equation moving the logs and constants and signs around and so finally you can solve for n as a function of time and you find out that it’s exponential in nature if you start with a number of objects n0 the number of objects left after a Time T is given by E Theus Lambda T times the original number of objects n0 now let’s talk about this constant of proportionality which we’ve been calling Lambda so in order to satisfy the requirement that the total argument of the exponential function be dimensionless it must be true that Lambda has has units of inverse time one over time or one/ seconds per second Hertz in the units of oscillatory phenomena it’s convenient to therefore Define Lambda as one over some characteristic time which I’ll denote with the lowercase Greek letter Tao where too is known as the time constant of the phenomenon well what does it actually mean for t to reach to the time constant well if you allow enough time to pass that one time constant’s worth of time goes by you find that 63.2% of the original number of objects are gone for unstable particles this characteristic time is what is known as the lifetime of the particle and you can actually show using some math we’ll develop later in the course that mathematically to is also equal to the average time that an unstable particle exists so it has two meanings one it’s the time after which 6 63.2% of the original n0 objects have disappeared from the system and two it’s the average time that any randomly picked unstable particle will exist now where does the halflife come into all this well you can show that the halflife of an unstable particle which we could denote as t with a subscript 1/2 is directly related to the time constant tow by the following simple equation the halflife is equal to the time constant time the natural log of 2 so when we say quote the muon has a lifetime of 2.2 micr seconds unquote we’re referring to the time at which there is a 63.2% chance that any single muon has decayed vanished gone away from the original sample of muons now let’s talk about muons and observing them and their origins in the world around us muons are not naturally occurring in the same sense that atoms are naturally occurring atoms are generally speaking stable they stick around for a long time and they form large structures because they have a chance to bind to each other through chemical means which is just electromagnetism and action muons on the other hand are a bit stranger you have to make them and because they don’t live very long you have a very limited opportunity to study them once they come into existence now thankfully nature does make them all the time and it does so because the Earth is constantly bombarded by particles from outer space that are smash ing into the atmosphere at very high speed very high kinetic energy and these things are known as cosmic rays and when cosmic rays slam into the Earth’s atmosphere they result in a whole bunch of particle interactions that ultimately spray muons down onto the Earth among other things so they do this by smashing into nitrogen or oxygen nuclei having all kinds of nuclear reactions in the process that produce produce a whole bunch of other particles and I’m not going to worry about what those are right now but ultimately muons can result from this and the symbol mu with a minus sign next to it denotes the muon with its natural negative electric charge there are also positively charged muons and that’s a subject we’ll come to later in the course now Anderson and neet Meer who originally discovered the muon did so using Showers of particles or from cosmic ray so-called cosmic ray showers and they did so by putting detectors at different altitudes in the Earth’s atmosphere so for instance they did a bunch of experiments with a detector located on top of Pike’s Peak which is 4.3 kilometers above sea level uh and they did a bunch of experiments at home base at Caltech in Pasadena California which is roughly at at sea level and it turns out in the decades we’ve been studying cosmic ray interactions and muon production we’ve learned that most of the muons that are produced by cosmic rays are made roughly at a height of 15 km above the Earth’s surface surface that’s not the top of the atmosphere but it does correspond to the place where the density of the atmosphere gets big enough that these interactions of cosmic rays and nitrogen and oxygen molecules or nuclei uh get very high in probability and so we get a lot of muons that get produced at at that part of the atmosphere now based on the known instability of the muon one might expect that if one counts a certain number of muons at a high altitude say counting a number N1 then by the math of unstable particle decay in using the known lifetime of the muon when it’s nearly at rest that is to for the MU is roughly 2.2 micr seconds one could accurately predict the number of muons you should expect to see at a lower altitude N2 now at that lower altitude because particle Decay has had a chance to happen we expect fewer total muons to be found if we make a 100 muons or a thousand muons at 15 kilm above the Earth’s atmosphere and we go down a bunch of of kilometers we don’t expect to find uh the same number of muons we expect to find typically fewer all right now this is very interesting here let me show you this so here in the basement of Fondren science the physics department has a small experiment set up that allows us to capture Neons created in cosmic ray showers above the Earth trap them by trapping them in in atoms and a material in this device over here on the left and then after trapping them we can wait and see how long they stick around until they Decay so all of this equipment is designed to establish the time uh at which an a muon is trapped and then the time at which it then subsequently decays because it doesn’t live forever and if we take a look at the data here what we find is that when we trap these muons and hold them nearly at rest in our reference frame indeed we see an exponential falloff in the number of muons that survive after a certain amount of time as predicted by the theory of particle Decay and we can see that after about 2.2 micros that there’s a roughly 60 to 70% chance that any single muon will have already decayed exactly as previous experiments have determined so this is our own little way of caging muons using atoms to trap them then waiting to see them Decay and measuring the time between those two events in a frame that’s essentially at rest with respect to the muon and indeed this is how we figure out for instance that uh the muon lifetime is about 2.2 microsc this experiment alone here in the time it’s been operating which looks to be about 2,300 hours or so has trapped and observed the decay of about 1.6 million muons so just think about the sheer number of muons that must be raining down on the surface of the Earth all the time we’re capturing just a tiny slice of all of those they’re fantastic laboratory for looking at the little clocks that fundamental particles carry around with them so that we can study time um using the tiniest building blocks of the universe now the mu’s short lifetime should radically cut down its numbers as we go lower and lower into the atmosphere and in fact the the effect is quite stunning so let’s imagine we give the muon the best possible chance of making it to a low atmospheric height so close to the surface of the Earth now to do that we’re going to crank its velocity up to the fastest that anything that we know of can travel and that’s the speed of light so we’re going to set the speed of a muon that’s just been produced at 15 km above the surface of the Earth we’re going to set its velocity aimed straight at the surface of the Earth to the maximum it can be 2998 x 10 8 m/s so if you crunch the numbers you’ll find out that in one lifetime a muon can travel just 66 km not even a kilometer it doesn’t even go 1 15th of the way way down closer to the surface of the Earth but at this point it’s already had a 63.2% chance of decaying there’s a 63.2% chance that that muon won’t exist anymore by this point but let’s imagine it survives and it makes it two lifetimes after two lifetimes at at the speed of light it could have gone 1.3 km doubling the distance it’s traveled into the Earth’s atmosphere and now having made it a little a little more than 1/5th of the way into the Earth’s atmosphere but by this point it pays the immense Penal of having a probability of 86.4% of having already decayed 10 lifetimes will only bring a muon 6.6 km into the atmosphere that still leaves it about 8 km above the surface of the Earth but by then it has had a 99.995% chance of of decaying there’s really very little chance that muon really makes it this far and if you take it twice as far 20 lifetimes the probability is even smaller so the bottom line is that we don’t really expect if we produce a th000 muons at 15 km to find really any of them down at sea level so what actually is observed well shown at the right is some data it’s real data taken from an experiment that really can count muons at different altitudes and the graph shows the number of counts per minute versus the altitude where the measurements were taken and these measurements were taken by High School teachers who were involved in a program called quarknet this program engages teachers in K through 12 typically High School teachers in real physics research environments and this data is actually taken from an experiment they did that was reported in the article that’s listed in the footnote on this slide now what they found was that if the experiment was run 3.5 km above the surface of the Earth above sea level they found about 300 counts per minute of muons at that height now let’s use the Galilean and Newtonian Assumption of time that time passes at the same rate for all observers that is whether the muon is moving or not its clock and clocks on the ground tick at the same rate now that’s a total violation of the assumptions of special relativity and of course the conclusions that one would then draw from the Loren transformation but we can make a prediction using the Newtonian or Galilean idea and so we can basically estimate how many counts per minute we expect at half a kilometer which is roughly the lowest height where the teachers took data now what You observe is that at 3.5 km the number of counts is about 300 per minute 300 muons per minute passing through the detector and if we give the best case chance of all those muons making it down to half a kilometer above the surface of the Earth we find out that um we we should expect the yield to go as e to Theus y over C * to where C is the speed of light and to is the lifetime and so after a height change of just 3 km going from 3 and 1/2 km to half a kilometer here we expect to find at most about 3.2 counts per minute from muons that are produced uh at this altitude of 3 and a half kilomet is that what we actually observe and the answer is heck no in fact the teachers observed not three counts per minute but 100 counts per minute at both sea level and about half a kilometer above the surface of the Earth it’s pretty hard to tell the difference between those two sets of data so why would that be why would it be that the Galilean Newtonian prediction or at least its assumption that time is the same for all observers regardless of the state of motion uh would not get this experiment right it seems so simple we know the lifetime of the muon when it’s at rest you know the height difference between where you make the first and second measurements you just do some counting should be easy right and you don’t even get close to the right answer so why would that be well I think we already know the answer the answer is time dilation special relativity with its Loren transformation that’s supposed to be valid for all speeds up to that of light will help us to understand this so let’s relate what’s going on in the muons ref reference frame which we’ll call S Prime and what’s going on in the earth’s reference frame which we’ll call S so we choose the earth and the atmosphere to be at rest we choose the muon to be moving uh so it’s viewed as a moving object with respect to the uh the Earth and so so we can call that the moving frame now in the reference frame of the muon where it thinks it’s at rest its lifetime is 2.2 micros recall that this is the lifetime as observed when the muan is nearly or exactly at rest and that would be its proper lifetime when it’s exactly at rest the proper lifetime is measured in the frame where all events happen at the same location in space for the muon coming into existence and going out of existence all happen at the same place itself and so that’s the frame in which proper lifetime is defined that’s also how we measure the lifetime of the muons we stop it and we let it decay and we see how long that takes typically and so that’s the 2.2 microsc associated with how long the muon lives now the Loren transformation would predict that the time measured by an observer on the earth the time that’s passing in the muons frame of reference will be different from a person who would measure the time but ride along with the muon thinking the muon is the thing at rest the whole time so we can take observations in the frame of the earth observations of say the clock ticks in the muons frame at T2 and T1 and take the difference the delta T between those ticks and we can relate those to the spatial coordinates where all the events happen and the time measurements as observed in the frame of the muon the S Prime frame we’re just using the Loren transformation one more time here now all the events in the muons frame of reference being created in the upper atmosphere decaying later at a time T2 where the Earth is closer to the muon they all happen in the same place in the muons reference frame in other words X2 Prime is equal to X1 Prime therefore this equation simplifies and the time difference in the earth rest frame is relatable to the time difference in the muons frame of reference by a factor of gamma times the time difference in the muons frame of reference now the lifetime of the muon in its frame of reference is 2.2 micros so delta T prime or T2 Prime minus T1 Prime is 2.2 micros so special relativity would predict that from the perspective of an observer on the ground the muon would appear to live longer than would be expected if it were at rest as well this is completely in accordance with the observational evidence more muons many more muons are obs deserved to survive to a lower altitude than would be expected from classical physics and its Assumption of the absolute passage of time for all observers so the data told us that of the say 300 muons per minute observed at 3 and a half kilom roughly speaking 100 per minute of those survive around half a kilometer above the earth’s surface in the reference frame of the earth we can relate these numbers to the observed Decay time of the muons in their rest frame that is TOA the proper time and also the lifetime of the particle and the distance that they travel from the perspective of the earth and atmosphere rest frame y as well as the typical speed of muons so what we find is that taking the decay equation n = n0 * e to Theus T over gamma to then tells us that we can solve for the velocity and the gamma factor of the muons using the data we we know n and we know n0 from the data um we know why because we know the height difference that the teachers made the measurements at we can solve for this quantity gamma V which is related of course ultimately to the speed of the muons in the atmosphere relative to the Earth so if we do the algebra here and solve for gamma V we get the following equation now I’m going to leave it as an exercise to the viewer um we want to solve ultimately for either Gamma or V but since there each a function of the other we have to do some algebra to isolate one or the other and to help you along with this recall that the gamma factor is defined as the uh 1 over the < TK of 1 – v^2 over c^2 and and that means that that the velocity if you solve for that is equal to C * the < TK 1 – 1/ gamma 2 and from this you can take gamma * V and you can get a nice expression for that so gamma V can actually be written entirely in terms of of gamma which is interesting and if you use that trick you can uh get to isolated expressions for either Gamma or V from from this equation here on the right hand side so go ahead and try that yourself as an exercise but you should find the following things you should find that the gamma factor for these muons assuming that all the 300 that are created at or appear at an altitude of 3 and a half kilm then could be counted or not at .5 km above the Earth’s surface and you then based on that assumption estimate that the gamma factor is around 4.3 and if you solve for the velocity of these muons they are radically close to the speed of light they are 2.91 x 10 E8 meters per second in speed relative to the Earth in the atmosphere now from the person on the ground’s perspective that Journey from three and a half kilm to half a kilometer above the surface of the Earth takes about 10.3 micros which is way longer than one lifetime of a muon so it’s no wonder muons make a fantastic and early laboratory for tests of of special relativity nature is readily making lots of them per second in the upper atmosphere they can be measured using technology that was available in the early 1900s at least the first half of the 1900s they can be observed and to see when they Decay and how often they Decay and so forth and all of that together can be used to assess the validity or not of special relativity and of course what we find is that special relativity wins the day it is the correct description of space and time for inertial reference frames and it’s remarkable how well it actually works now of course the atmosphere is complicated the production of muons in the atmosphere is complicated if you really wanted to do a super thorough job of this you would have to do a detailed simulation of the interactions of cosmic rays in the Earth’s atmosphere producing muons at various Heights and then see how many you count at various Heights with and without special relativity if you do that we find that with special relativity we can exact produce the atmospheric data without special relativity we utterly fail to reproduce the atmospheric data it really is true that special relativity is the correct description of space and time and motion now I find it’s instructive to quickly take a look at this same situation but from the muons perspective that is if you could ride along with the muon at its ridiculous speed what Would You observe of course the mu’s frame of reference it’s at rest and the earth and the atmosphere are rushing toward it or in the case of the atmosphere past it so you and the muon come into existence very suddenly 3 kilomet above the final Earth observation place now that’s in the perspective that 3 km statement that’s made in the frame of an observer on the earth we’ll get to the distances in a second in the muon frame what you can say for sure as you come into existence the Earth is is far from you it’s racing toward you at a speed of v and it’s getting closer to you all the time and at some point you’ll go out of existence and the question is how far is the Earth and atmosphere going to move in the time between those two events coming into existence and going out of existence so the perspective of the earth Observer is on the left in this cartoon and the perspective of the muon Observer is shown on the right in this cartoon we don’t know the distance between the surface of the Earth and the muon in this picture we only know it from the original experiment in the earth rest frame but here we are confident that the Earth is rushing toward us at the opposite velocity that’s measured in the earth frame for the muon heading toward the earth now from the muons perspective of course it’s standing still in all events coming into existence and decaying they happen at the same location in its frame of reference therefore the time it typically is going to stick around is going to be 2.2 microc in its frame of reference it sees the earth below it when it comes into existence and it sees that that surface of the Earth rushing toward it at at a velocity of negative V so how far does the muon have to go to make it to its destination from its perspective and its reference frame now time is ticking away at whatever rate it goes at for the muon and ultimately it can measure time using its own lifetime which is about 2.2 microseconds nothing funny with time in its rest frame but of course the distance where the muon was created above the surface of the Earth is 3 km in the rest frame of the earth that’s the frame where the earth and its atmosphere appear to be at rest and that makes that distance 3 kmers the proper length or proper height above the surface of the Earth that is the longest distance that any reference frame would measure between where the muon is created and say the surface of the Earth the the muon on the other hand will see the earth atmosphere system as moving and therefore distances in that system contracted along the direction of flight and the length or height above the surface of the Earth that it will measure will be the proper length 3 km divided by gamma and this comes out to be about 23.3% of the proper length or .699 km 699 M that’s the distance that the muan perceiv between where it comes into existence and that final measuring point which was 3 km away in the frame of the Earth in the atmosphere so from the muons perspective we conclude that it observes that the distance it will travel is contracted compared to what observers on the Earth are seeing and that contraction factor is one over gamma from the mu’s perspective the distance between the place in the atmosphere where it came into existence and where It ultimately decays is greatly shortened requiring only a time of delta T Prime of about 2.4 micros to make the the the trip because the earth atmosphere system is contracted and moving relative to it and so you know given that it lives about 2.2 microsc in its reference frame it’s absolutely plausible that it could make it that full distance that people on Earth said it went it’s just that the people on Earth are confused because the distance is shorter than they claim from the muons perspective so while observers on Earth and an observer moving with the muon would disagree on the reason for the muon reaching the lower measurement point they all agree that it’s very likely to happen the Earth Observer argues that the reason it makes it is because time in the muons frame is passing more slowly than they claim because the muon is moving and so that takes longer to Decay as a result of that it’s able to cross the 3 km Gap even though it should have only live 2.2 micros because time has slowed down for it while it’s in motion the muon obser Observer says no our clocks are working just fine what’s going on is that because the Earth and atmosphere are rushing toward us they’re in motion and so they seem contracted along the direction of motion and as a result of that we don’t have to go that far to make it to say the surface of the Earth and we’re definitely going to make it in about 2.2 microseconds or so that’s why we made it so far now they’re both right even if they have different reasons for what happens they both observe the same outcome the muon makes it to the surface of the Earth but they disagree on the space and time reasons for that and that’s okay because the Lauren’s transformation allows them to relate their perspectives and put their measurements into the other person’s frame to see what’s going on and resolves the Paradox in that sense so to review in this lecture we have learned first of all what is a muon it’s a subatomic particle it’s about 200 times heavier than the El electron it’s about five times lighter than the proton and it has the same Elementary charge as the electron so it’s its own thing and it would take decades after it was originally discovered to finally fit it into the the the sort of final picture of nature that we’ve reached today the muon regardless of what it really is is an outstanding laboratory for testing predictions that are made using the Loren transformation specifically about whether or not muons given their very short lifetimes should be able to travel the vast distances from where they’re created in the Earth’s atmosphere to where they can be measured down on the surface of the Earth and in fact we find that muons in vast numbers make it from where they’re produced in the upper atmosphere to the ground but they’re not supposed to if time passes at the same rate for all observers and all frames of reference so it may seem weird that time doesn’t pass at the same rate when you’re moving but it’s the truth we have direct tests of this not only with muons but with many other systems as well and in many ways the muon wound up being the very first direct test of the validity of special relativity and it it held up against that that test beautifully to live another day and make more predictions which is what makes it such a spectacular theory of space and [Music] time in this lecture we will learn the following things we’ll Begin by learning what is the classical Doppler effect on an oscillatory phenomenon like a wave we’ll also learn about the effect of the motion of a light source on the characteristics of the light other than its speed and finally we will learn how to compute the so-called special relativistic Doppler effect on light and interpret the effect on observations of the world around us let’s begin by recalling oscillatory phenomena from introductory physics specifically let’s look back at something called simple harmonic motion this is a kind of repetitious motion that has a time and space structure that allows itself to be described using S or cosine functions of space and time so for example depicted in the graph at the bottom of the slide we have the vertical position of some object as a function of the horizontal position of the object and we see that the vertical position varies gently upward then downward then upward again and then repeats with the horizontal position and this motion of the y coordinate with with respect to the x coordinate lends itself to a direct description in this case using a sign function now a wave phenomenon such as a water wave or a sound wave can similarly be described using exactly this kind of mathematics a sound wave is a region of high compressed air followed by low compressed air followed by high compressed air and so forth It’s a so-called density wave in air a water wave is similar L an increase in the number of water molecules in one region of water and a decrease in the number of water molecules in another a rising and a falling of the surface of the water these wave phenomena are oscillatory in nature and can be exactly described using the same kind of s or cosine function approach that we apply on simple harmonic motion now the wave phenomena just like oscillatory phenomena have characteristic numbers that describe their spatial distribution there’s no one place where a wave is and where it is not for instance you could say that there’s more of the wave in this region of Y and less of the wave in this region of Y the wave is a structure that’s spread out in space and it has both a spatial structure and because it can move uh in time it has a Time structure as well we have to use use

certain quantities to characterize the overall macroscopic shape of an oscillatory phenomenon or a wave and the wavelength denoted by the lowercase Greek letter Lambda is one such number for instance the wavelength of a wave like the one depicted here could be taken as the distance between crests of the wave the locations of the Maxima the maximum displacement from zero in the y direction or it could be taken as the zero displacement of the phenomena really picking any two similar points on the wave and drawing a line between those points horizontally will give you the wavelength now if we were to observe this phenomenon passing Us by by picking a reference point in space and just watching it Go by that point the time between Maxima or Minima passing the same spatial reference point is known as the period capital T of the wave this is the time between the same thing happening over and over and over again in the wave phenomenon now the inverse of the period one overt is the rate at which for instance Maxima passed that point and it’s known as the frequency and it can be denoted in one of a couple of ways for instance the lowercase letter f for frequency which equals 1 /t or it can be denoted using the Greek letter new which looks like a little curved V that’s also used often to describe the frequency of a wave and again that’s just equal to one over the period or 1/t frequency have units of per second or Hertz h r TZ the unit of frequency now the speed with which waves move in space during some unit of time is actually given by a very simple product of frequency and wavelength if you want to know the speed of a wave you just take the wavelength Lambda and you multiply it times the frequency f and of course for a light wave the speed with which all light waves move is known to be C the speed of light and so this will just be again the product of the wavelength of the light wave and the frequency of the light wave but that product will always yield C 2998 * 108 m/s regardless of whether an observer is in motion relative to the source of the light or not we know that already as one of the postulates of special relativity now summarizing again the gross properties of waves it’s helpful to pick a characteristic point on a wave and think about the repetition of those characteristic points as representative of the spatial or temporal distribution of the wave phenomenon so for instance we can think of waves of sound or waves of light as merely being represented by lines or planes so for instance uh the location of a line in two Dimensions or a plane in three dimensions could indicate a location in space of a maximum of the traveling wave using this picture at the right you might imagine that each of the locations in space marked by one of these red planes is a place where you would find a maximum of the wave having been sliced through by the plane this is a very common way to quickly and simply sketch a wave without having to draw the s or cosine function the distance between lines or planes is the wavelength that’s the sort of cartoon way you represent that particular feature of a wave in the image now such a line or plane would be referred to as a wave front and wave fronts can be used to characterize the location and space of a particular point on a wave and all of the repetitions of that point the frequency of such a phenomenon can be thought of as how many fronts per second are emitted by the source so if you think about one plane and then another plane and then a third plane being emitted by the source the distance and time between those planer emissions but would be the period of the wave and one over that would be the frequency the rate at which it emits wavefronts now this brings us to the so-called classical Doppler effect and I’m going to use sound waves to motivate this because most of you have probably at one point in your life or another actually experienced the Doppler effect with sound waves the Doppler effect occurs when the source emitting a wave is itself moving relative to an observer so if we’re talking about a sound wve here we’re talking about a listener someone who can receive the pressure changes in their ears and when that Observer is moving relative to the source the Doppler effect can occur and this is actually Illustrated in this cartoonish animation below a car starts emitting sound perhaps by the driver laying on the horn and the wave fronts are represented by those red circles so everywhere you can locate a point on a red circle would be a wave front of the sound waves and they’re emitted at rest uniformly in all directions but as the car accelerates forward the wave fronts in front of the car begin to pile up the sound waves get closer together the wavelength shrinks and behind the car we see the wave fronts spread out the wavelength gets bigger so in this example using sound waves we have the effect that in the direction of motion of the emitter the car honking its horn for instance and ahead of the Source the wave fronts are pressed together more densely shrinking the wavelength and thus increasing the frequency with which waves will strike our ears if we were to be standing on the backs side of this moving object while it’s moving away from us sort of against its direction of motion the wave fronts are more widely spread apart than they would normally be and this increases the wavelength and thus decreases the frequency with which these waves reach our ears so to human ears it’s the frequency of waves that determines what we call pitch high pitched sounds are also high frequency sounds the wavefronts are striking our ears more frequently and vice versa low pitched sounds are low frequency sounds the time between wave fronts hitting our ears is [Music] longer [Applause] [Music] now let’s think about the Doppler effect on light waves there is a classical Doppler effect on light waves but because time and space measurements are also relative to the frame in which you’re making the measurement there is a Rel relistic component that gets added to this kind of pitch shift even for light waves so it’s true that while all observers may agree on the speed of light we know that special relativity leads to the conclusion that space and time measurements May differ between observers in different reference frames now wavelength is a space measurement and frequency is a Time measurement so couched in that language it should come as no surprise then that while observers in relative motion all agree that a wave of light travels at C 2998 * 108 m/s regardless of what frame of reference they’re in observers in different frames will disagree on the wavelength and frequency of that light wave now to measure wavelength for example is to be able to simultaneously locate Ma Maxima on a wave think back to our discussion of measurement and how one measures distances on a moving object there are different ways you can do it but one of those ways is to simultaneously collocate points on the object in this case one maximum and then the next maximum but we know that simultaneity is a frame dependent statement and in moving frames of reference we know that the objects pinned to those frames appear contracted along the direction of motion from observers that are not in that frame and are for instance at rest with respect to that frame similarly to measure frequency is to be able to measure the time displacement between two events at the same location in space how often or how much time happens between wavefronts going by a single point and we know that from the perspective of a frame that’s at rest the time in the moving frame passes more slowly and these two frames would disagree on the amount of time between two wavefronts so the combination of the classical piling up of wavefronts or stretching out of wavefronts due to the motion of the source with these time or space effects that come from the special relativity postulates and the Loren transformation come together in what is known as the special relativistic Doppler effect and we will derive it here using the transformation applied on top of the classical Doppler effect calculation and we will discuss the implications of this phenomenon for observing the universe it has some extremely deep impacts on our ability to understand nature even distant parts of the universe where we have no physical access to moving objects so to motivate the derivation of the relativistic Doppler effect I’m going to start by talking about just the sort of classical Doppler effect and to do this what I want to do is have you imagine a light emitting device Illustrated here as a blue ball that is sending out wave fronts to the left along this coordinate axis and it’s doing so at regular intervals in its rest frame so for instance we might have a moment in time T1 Prime in in the frame of the source where it emits its first wavefront its first Maximum is emitted from a point say for instance 14 along this coordinate axis and then at a later time T2 prime it emits the second front so in the time between emitting the first front and the second front of course the first front has moved at the speed of light to the left and it’s now at 13 on the x-axis at the moment T2 Prime that the second front is emitted by the source and if these fronts are emitted regularly as would be true in a simple harmonic oscillatory or wave phenomenon then there will be a third front emitted at T3 Prime and a fourth front emitted at T4 Prime and the distance in time between these successive emissions of fronts will all be the same corresponding to the period of The Source Capital T Prime in the frame of the emitting source so T2 primeus T1 Prime will be the same as T3 Prime minus T2 Prime all of those intervals between neighboring wavefront emissions will be the period of the source as observed in the frame of the source this defines the period of the light wave thus we have a regular frequency in the source frame we can write that frequency in the source frame as F Prime = 1 / T Prime now that was done with the source sitting at rest along this coordinate axis and so in that P picture an observer sitting at zero on the coordinate axis would agree that the wave fronts all arrive with a time between them equal to what the source said it would be because the source and the Observer were at rest with respect to each other but now let’s imagine that the light emitting device that’s sending out those wave fronts at regular intervals and its own reference frame is moving with respect to the Observer at zero in the above coordinate axis so let the velocity of the source be plus v that is it’s moving away from the origin to the right and entirely along the X AIS treat The Observer is being at rest we’ll call that frame s and the source as being in motion we’ll call that frame S Prime let’s think about what will be the distance between wavefronts arriving at The Observer from the perspective of the source so we’re doing all of this from inside the source frame S Prime we will transform to the rest frame of the Observer later so here is our source it’s at location 12 along the x axis and it emits its first front front one at time T1 Prime and the wavefront moves toward the zero point on the X x axis at the speed of light C now the next time that the source emits a wavefront it has itelf move to the right from 12 to3 in the meantime the light wave front that it emitted the first front front one has moved at the speed of light to the left and in this particular example it winds up being at 10 on the coordinate axis so in the frame of reference of the source the distance between the wave fronts would have been Lambda Prime the wavelength and that’s depicted here on this cartoon showing where the original unstretched wavelength of this phenomenon would have been if the source had been at rest the source had been at 12 when it emitted front one front one is now at 10 and so the distance between those two would be Lambda Prime the wavelength of the phenomenon in the frame of the source but the source is now moved and so it’s at this location 13 that it emits its second front so by the time time emits the second wavefront at time T2 Prime the first wavefront is a distance of C * T2 primeus T1 prime or c times the period in the frame of the source and that’s just equal to Lambda Prime that’s the distance from where it was emitted but the source is now farther from the Observer by an amount of V * T2 Prime minus T1 prime or V * the period when it emits the second wavefront and again that’s depicted up here so this is the distance from the point of emission that front one has traveled which technically would have been one wavelength of the original emission but the source has moved back uh further along the x axis by a distance VT Prime V * the period and so the source will argue that as a result of this an observer who is looking at these wavefronts coming at them sitting at point zero should see the combined distance of Lambda prime plus VT Prime between the two wavefronts and and this will actually be the observed wavelength of the phenomenon according to the person uh riding with the source that will be what the Observer sitting at zero should see so this is all Illustrated above we take Lambda Prime and we add VT Prime and that’s going to give us the observed wavelength in the frame of the source so this is all Illustrated and we can write the equation down adding these two together and then we can rewrite this in terms of frequencies by remembering first of all that period is equal to 1 over frequency or frequency is equal to one over period and that the speed of light is equal to the product of wavelength and frequency similarly the wavelength uh for instance in the frame of the source will be the speed of light divided by the frequency in the frame of the source and the wavelength according to the Observer from the perspective of frame S Prime will be equal to the speed of light divided by the frequency according to the observ in that frame so we then find again all of this from the perspective of frame S Prime that the observed period at at zero should be 1 + V / C all divided by the frequency of emission fime so maybe pause here and try to work all this out for yourself but again keep in mind that we have not yet transformed this observation into the rest frame that is the frame in which the Observer is at rest right now we’re calculating The observed wavelength or period according to what the Observer should see if their uh time measurements were absolutely in agreement with the moving Source there’s this is the classical Doppler effect the stretching or compressing of wavelength with the motion of the source we haven’t yet included for instance time dilation or length contraction in all of this now we’re going to take that last step and to Aid Us in notation here we’re going to begin by defining a very convenient symbol and that is the lowercase Greek letter beta which is rather regularly used to represent the ratio of the speed of the frame for instance divided by the speed of light because speeds V never exceed the speed of light C and because speeds can never be any less than zero beta is a number that goes from zero to one zero for things that are at rest one for things that are moving exactly at the speed of light and can take all values in between in terms of beta the claimed period of the phenomenon as observed by The Observer in frame S Prime should be 1 plus beta divided by the frequency of emission but again so far all of this is in the S Prime frame this is what a person in S Prime follow following along with the source would argue is what the Observer should see the original frequency of emission from the perspective of the source f-prime which we can just call F with the subscript source and includes the relative speed of the source in The Observer V and the period and frequency with which a person in the source frame would expect the Observer to receive the wavefronts T Prime observed or you know correspondingly the observed frequency however if we now do the special relativity and use the Loren transformation and transform this stuff into the actual frame of reference of the Observer we know that there’s going to be another effect here and for instance we could summarize that by saying that it will be the relativity of time time in the source frame where all the missions happen at the same location is proper time the source always says that wavefronts are being emitted from its location in space and as a result of that that’s the frame where proper time will be observed but in any other frame moving relative to the source time dilation will be what is observed that is the passage of time on the moving object will appear to be slower than the observers on the moving object would claim and it’s given simply by taking the proper time and multiplying it by gamma so the time the period observed in the rest frame of the Observer will be gamma time s the period that the source claims The Observer should have seen according to the classical Doppler effect so we then finally arrive at a expression for the period of the phenomenon of the light wave as observed in the rest frame so we start by just saying that the period observed in the rest frame will be equal to gamma times the period that should have been observed from the perspective of the moving frame T Prime with the subscript obs we can substitute in with 1+ beta over the source frequency fime and we can do some algebraic gymnastics to sort of rewrite this in a more pleasant looking form we’ve got gamma and we’ve got beta of course gamma depends on beta gamma has v over C all squared inside of it that’s beta squared so it’s nice to try to rewrite this all either in terms of just beta or just gamma and so with a little algebraic gymnastics starting with writing gamma is 1 over the < TK of 1 – beta 2 you can then do a little work and show that a final neat looking expression for this is that the period observed in the rest frame is equal to the square root of the ratio of 1 + beta over 1 minus beta all times 1 over the source frequency the frequency of emission from the perspective of the source itself in its rest frame so we can transform this into an observation of course of the frequency in the rest frame by simply doing one over T observed and that just flips the stuff in the square root upside down and you wind up with this neat little relationship that the frequency of the light observed in the frame that’s not moving we’ll see the frequency uh as emitted in the source frame where the source is at rest shifted by an amount given by the square otk of 1 minus beta over 1+ beta so we’ve solved the problem now we’ve derived the special relativistic Doppler effect the shifting of the frequency due to relative motion between the source of emission and the Observer of the light by considering the situation where the source is moving away from The Observer now this special relativistic doler effect is a combination of two effects the classical Doppler effect of just the effect effect of the moving source that adds extra space between the wavefronts but in addition to that the dilation of time due to relative motion of the source and the Observer proper time and therefore proper frequency would be in the frame of the source this is modified by a gamma factor to go into any other reference frame so the special relativistic Doppler effect is a combination of the classical Doppler effect with the relativity of space and time measurements and you actually would expect from just Newtonian and Galilean relativity that there’s a Doppler effect on frequency and wavelength of light but the special relativistic addition to that actually makes the effect even more extreme than expected from Newtonian and Galilean mechanics and in fact what we see in the universe is what is predicted by special relativity and not just the old mechanical Galilean and Newtonian approach to motion now now for a source that’s moving toward an observer that is approaching the Observer while emitting wave fronts the sign of the Velocity is all that needs to be changed we go from having beta the velocity moving away to negative beta the velocity now moving toward the Observer and in fact you can do the work yourself but this formula up here for the source moving away from The Observer can be transformed into the case for the source moving toward the Observer by flipping the sign of beta so taking beta and turning it into negative beta and all that does is it takes the stuff under the square root and flips it upside down so now we have the square root of 1 plus beta divided 1 minus beta that whole thing times the frequency of emission in the source rest frame so I would recommend you practice this calculation by checking for yourself that this second equation for an approaching source is correct um but once you’ve convinced yourself of that the shortcut is a really easy thing to remember if you can remember one of these two formulas you can get the other one simply by changing the sign of beta not too bad so let’s talk about some expectations from the special relativistic Doppler shift for example if a light source is moving away from us or toward us what do we expect to happen to the frequency of its light so for a source that’s moving away from us at speed beta along our line of sight we expect to scale The Source frequency by the following quantity the square root of the ratio of 1 – beta over 1+ beta now if you play around with this a little bit you’ll notice that this thing is always less than or equal to one it’s exactly one when beta is zero and if beta is anything other than zero its value decreases from one the frequency therefore that we should observe should always be lower than in the source’s frame of reference owing to the stretching of its wavefronts combined with the dilation of time now because as I said beta is a number that’s inclusively bounded between 0 and one we are taking the ratio of a number less than one and a number greater than one for beta that’s anything other than zero now if instead the source and observer are moving toward each other then we scale The Source frequency by this quantity the square OT of 1+ beta / 1us beta and again if you play around with this you’ll find out that this is either always equal to one or greater than one this means that the observed frequency is always greater than what what is observed in the frame of the source since we’re dividing a number greater than or equal to one by a number that’s less than or equal to one so to summarize all of this for a source that’s moving away from The Observer of the light the frequency that the Observer sees will be lower than the frequency that’s observed in the rest frame of the source itself and similarly for a source that’s moving toward an observer The Observer will always see that the frequency is increased over the frequency as observed in the rest frame of the source of the light emissions these equations are all for frequency but we can very quickly derive the equations for wavelength using the fact that the speed of light is equal to wavelength time frequency so if we go through the brief algebra gymnastics for this we find that we get the following equations for The observed wavelength depending on whether the source is moving away from The Observer or if the source is moving toward an observer so as expected when a receding Source uh is in is present this gives us a lower frequency and a longer or greater wavelength so the frequency has gone down therefore the wavefronts are farther apart from each other because the wave is still traveling at the same speed C on the other hand when the source is approaching us we get the higher frequency which means the wavefronts are coming at us more often and that means a shorter wavelength will be observed for the phenomenon so let’s talk about the perception of light color due to the full relativistic Doppler shift so I’ve Rewritten here the equations for the weight wavelength that an observer sees depending on whether the source is moving away from the observer in which case the wavelength is stretched by the motion or if the source is moving toward an observer in which case the wavelength is compressed uh by the motion receding sources of light are said to Red shift compared to when they are at rest and that’s because longer wavelengths correspond to redder light than shorter wavelengths which correspond to Bluer light I’ve Illustrated this over here on the right using a spectrum and specifically I’ve isolated the visible or color spectrum of the electromagnetic frequency spectrum so for instance red light near the edge of where the human eye can detect the color red has a length of about 700 nanometers or 700 billions of a meter blue light or uh violet light which is at the other end of our ability to see comes in at around 400 nanometers or 400 billionths of a meter in length blue light has a shorter wavelength and thus a higher frequency than red light so if a source is moving away from us the wave fronts get stretched out and that would take something that’s Bluer and shift it more toward the red end of the light spectrum and conversely an approaching source is said to be blue shifted because this results in shorter wavelengths which corresponds to the Bluer end of the color spectrum now of course it’s possible that if you have an object that’s already say Violet in color and it’s moving toward you very rapidly at a significant fraction of the speed of light the shifting effect can be such that it actually shifts uh so blue that it goes outside the visible spectrum and then you’d have to look for it in ultraviolet or x-rays or other similar very short wavelength electromagnetic phenomena uh similarly if an object is already very red and on the edge of the ability of the human eye to see it and the source of the light is moving away from you appreciably quickly this can result in a red shift that puts it into the infrared or even microwave or radio depending on the speed of the object that’s emitting that light you can imagine therefore that this has some strong implications for measuring our place in the whole Cosmos for example without making physical contact with distant stars or galaxies which are collections of billions or trillions of stars it’s possible to actually determine whether or not those objects are receding away from Earth or approaching toward Earth based on the degree of the color shift of their atomic Spectra let’s take a look at an example of this through long centuries of observation of distant objects by astronomers and especially by breaking down the light from distant objects into their component colors the so-called color spectrum or atomic Spectrum astronomers have determined that the stuff that makes up everything out there is the same stuff that makes up everything down here on Earth and that is at least for the Luminous stuff the stuff that emits light or absorbs light that stuff is atoms and the atoms that are out there appear to be the same as the atoms that are down here on Earth iron has the same atomic spectrum whether it’s found on Earth or in the heart of of a star so as a result of that we can look at the light coming from distant objects figure out what atoms it’s made from and knowing the pattern of light each atom gives off determine whether or not first of all it’s composed of certain atoms and second of all whether it’s moving relative to us so here’s how you figure out the motion the Spectrum on the left over here on the slide is actually from our own star the sun the sun is not appreciably getting closer or farther away from us over the course of a day or a year we’re going around the Sun and our distance is changing slightly as we orbit the Sun every year but it’s not happening so fast that you get an appreciably different shift at least to the eye in the color of the Sun so we can consider the sun to be an atomic spectrum that represents a star at rest on the other hand on the right hand side is an atomic spectrum from a very distant so-called supercluster of galaxies that’s a cluster of galaxies of stars which are themselves uh clusters of stars and it’s named bas1 it’s not so important what it’s called but if you stare at this for a few moments you’ll notice that there’s an interesting similarity in the pattern of the light between our sun and the light that’s coming from all the stars that make up this distant supercluster of galaxies there’s a missing color line here in our sun and then there’s a a gap where there’s lots of colors and then there’s another missing line and then there’s another Gap and then there’s a missing line and then there’s a small Gap and there are two missing lines and if you look over on the right at the light from the super cluster you see that oh look there’s a line in the red that’s missing and then there’s that same Gap and then there’s a line in the yellow that’s missing and then there’s that same Gap and then there’s a line in the green that’s missing and then there’s that same small Gap and then the two dark lines that are next to each other it’s as if somebody took the pattern in our sun and shifted it toward the red end of the spectrum and this is exactly what we would expect from special relativistic Doppler effect if the supercluster is moving away from us thus stretching its emitted wavelengths of light longer from our perspective so these black gaps so-called absorption lines in the spect have the same pattern but in a slightly shifted location in the sun compared to this super cluster of galaxies and the fact that those missing colors are red shifted means the Galaxy supercluster is moving away from us that’s the conclusion from the special relativistic Doppler effect and you can actually then use measurement differences of the wavelengths between where the missing wavelengths are present in the Sun and where they’re present in the galaxy supercluster and using some astronomy you can actually estimate the relative velocity beta equal V / C with which the supercluster is receding from us this is incredible this allows us to measure velocities without having physical access to a material object all we have to do is look at the pattern of light that comes from its atoms and knowing that those patterns are the same patterns that should be found here in the atoms on earth look at the shifting of those patterns to determine the relative velocity of o to the distant object this kind of measurement is actually how we know that the universe as a whole is expanding so far as we’ve been able to determine all distant objects appear to be receding away from the earth as if carried along on the momentum of an initial explosion that set the whole universe in motion with all points expanding outward from every other point this implies the universe as a whole and on the largest distance scale is expanding with time so let’s review what we have learned in this lecture we’ve looked at the classical Doppler effect both in a cartoonish way and using the example of a moving Source emitting wavefronts along a horizontal axis we’ve then considered the effect of the motion of a light source where observers all agree that the light waves are moving at the same speed we’ve looked at the effect of the motion on characteristics of light other than that speed which isn’t changing the wavelength and the frequency of that light and by combining the classical Doppler shift with time dilation we’ve seen how to compute the special relativistic Doppler effect on light and we’ve even looked at ways that you can interpret that effect on observations and take observations of the natural world and use those to infer relative velocities on the grandest scales of the the [Music] cosmos third party another object from the perspective of those two frames and thinking about the velocity of that object as perceived in the two frames we’ll also learn how to properly add velocity of objects to frame velocities in special relativity let’s use a concrete example to motivate a kind of basic problem we can use going forward to think about the question of object velocities relative to moving frames of reference so the example I will pick for this is a non-copyright violating space wars recently in a globular cluster fairly nearby two ships were engaged in a chase the lead ship is moving away from the pursuing ship at a velocity given by the vector v the pursuing ship fires a projectile straight at the lead ship along the line of motion and at a velocity Vector U relative to the firing ship with what velocity does the lead ship observe the projectile to move now I’ve Illustrated this with a little graphic cartoon right here here we have the pursuing ship on the left the projectile it’s fired with the velocity of the projectile from its perspective drawn here in red the ship it’s pursuing over here on the right and the velocity of that ship being pursued relative to the the pursuing ship given by the vector v now in the Galilean or Newtonian view of space and time the answer to the question with what velocity does the lead ship observe the projectile to move is rather straightforward The observed velocity in its frame of reference U Vector Prime would be equal to the velocity of that ship with respect to the pursuing ship minus the velocity of the projectile with respect to the pursuing ship that would also turn out to be completely wrong when the velocities in this question approach velocities near that of light so for for instance if the projectile is actually a beam of light imagine a laser beam a laser cannon mounted on the front of the pursuing ship it turns on the cannon the beam is emitted this is a beam of pure light it should move at the speed of light if we plug that into this calculation we get all kinds of wrong answers here the lead ship doesn’t sh doesn’t see the laser beam approaching at the speed of light and we know that’s just not consistent with observation as encoded in the postulates of special relativity so what then is the correct addition of velocities in a problem like this and that’s the question we want to figure out in this lecture we can begin by writing down the Loren transformation equations treating the pursuing ship as the rest frame the lead ship as the moving frame and the projectile as an object to be located or studied in either frame the SpaceTime coordinates of that object in each frame are given as follows for example if we have the SpaceTime coordinates x and t in the rest frame we can get the space-time coordinates in the moving frame the S Prime frame using this version of the Loren transformation equations yielding X Prime and T Prime the location and the time at which the Lo is observed for the projectile in the perspective of the lead ship now we can write differentials of space and time using calculus DX Prime and DT Prime and this will allow us to work toward obtaining equations with velocities so for instance U Prime is the first derivative of x Prime with respect to T Prime after all that would be the velocity of the object as observed in the lead ship or moving frame U would be the first derivative of x with respect to T that’s the perspective of the projectile’s Velocity from the rest frame or the pursuing ship now if this particular step feels weirdly familiar to you in an earlier lecture I walked you through a brief example as to why the Loren transformation needs to be a linear transformation between moving frame coordinates and rest frame coordinates and we came dangerously close in that lecture to deriving the velocity transform albeit I was doing that for arbitrary Powers x to the n and T to the M for instance here of course it’s purely linear because it’s based on the Loren transformation and so if some of this feels awkwardly familiar you may flip back to the earlier lecture on the Loren transformation and have a look and see where the roots of this were planted so the differential of space in the moving frame DX Prime is going to be equal to gamma time the quantity dxus vdt and the differential of time in the moving frame is going to be equal to gamma * the quantity V / c^2 DX plus DT now we can take the ratio of DX Prime over DT Prime and this allows us to get the velocity U Prime of the projectile as observed in the moving frame or the frame of the lead ship substituting in with our differentials for DX Prime and DT Prime we arrive at this rather unpleasant looking equation but one of the nice things about this is that the leading gamma factors the 1 / < TK 1 – V ^2 over C sared terms they cancel out in both the numerator and the denominator and this leaves us with an equation that looks like this just in terms of the remaining differentials of DX and DT now if we divide the top and the Bottom by DT the little unit of time that we’re considering then we wind up with terms that go like DX over DT which is just U the velocity of the projectile in this case entirely along the x-axis and so this equation takes the following form which at the end of things doesn’t look horrible the velocity of the projectile as observed in the moving frame the frame of the lead ship is simply given by the velocity of the projectile as launched from the perspective of the rest frame the pursuing ship minus the velocity of the frame so the velocity difference between the lead ship and the pursuing ship divided by a quantity that goes like the Motion 1 minus UV over c^2 so we have arrived at a formula for combining the velocities of the moving frame with the velocity of the projectile as observed in the rest frame to allow us to compute The observed velocity of the projectile in the moving frame this equation is a substitution for the old galile and transformation addition of velocities equation and is correct from the perspective of special relativity so let’s plug in some numbers and see what we learn about projectile motion in the case where objects are also in relative motion to each other and observing that projectile as it moves and let’s begin by picking a low velocity situation where the ships are not really moving apart from each other all that fast I’ve decided to pick the lead ship having a velocity of just 1% the speed of light or 0.01 C and I’ve picked a projectile velocity that’s just three times bigger than that or 3% the speed of light 03c from the perspective of the firing ship the pursuing ship now from the above equation we learn that the lead ship observes the projectile approaching it at a speed of 0.02 C now if you stare at this for a moment and recall the Galilean velocity transformation you’ll note that this is exactly what we would have expected from the low speeed case where all the velocities of objects in the problem are are not really a large fraction of the speed of light although I’ve allowed them in this case to go up to a few perent the speed of light we actually get back exactly what would have been told To Us by the velocity transformation in Newtonian SL Galilean relativity that is that U Prime equals uus V now that doesn’t mean that this is exactly true at every decimal place there’s some decimal place where the Newtonian Galilean approximation uh to space and time and motion breaks down compared to the more accurate special relativistic calculation so let’s instead pick some bigger velocities let’s now assume that the lead ship is racing away from the pursuing ship at half the speed of light and then from the perspective of the pursuing ship it fires this projectile at 8/10 the speed of light8 C plugging those numbers in we find out that the lead ship observes the projectile to approach it at 1/ half the speed of light and if you stare at that again for a moment play around with the numbers on your own you’ll very quickly realize that this is definitely not what would have been predicted by the Newtonian or Galilean approach it’s not simply U minus V in this case now interestingly we can look at the case of when the lead ship is flying toward the pursuer so now we turn the lead ship around and we aim it back at the pursuing ship and flip its velocity Vector so that it’s moving at negative .5 C from compared to its original direction of motion in that case we see that the lead ship that’s now racing toward the projectile that’s been fired at it doesn’t observe that projectile to be moving in excess of the speed of light rather it observes it to be moving at 93% the speed of light and that’s again a distinction from what the Newtonian or Galilean approach would have yielded the old relativistic approach from Galilean relativity would have predicted that the lead ship observes the projectile to be approaching at a speed that is far in excess of the speed of light but we also know from the postulates of special relativity that one consequence is that nothing can move faster than the speed of light and so we see that that’s preserved here in the velocity transformation although the velocity of the ship is now aimed back at its pursuer and although the naive addition of velocities would give you something in excess of the speed of light the naive addition is not consistent with observ obervations of space and time and the speed of light and using the special relativistic transformation we see that while it’s true that the velocity of the projectile does appear to be larger than when the lead ship was racing away from it it does not exceed the speed of light but comes in at a pretty pretty fair fraction of the speed of light so let’s summarize what we’ve learned about adding velocities in special relativity keeping in mind that the cases that I’ve built these equations from All In involved an object velocity that was parallel or anti-parallel to the velocity of the frames if you have the velocity of the object in the rest frame and want to determine it in the moving frame then the left equation is what you want if on the other hand you have the object velocity in the moving frame and you want to determine it in the rest frame all that should change between the left equation and its corresponding equation on the right should be that you swap U and U Prime and you flip the sign of all terms that involve V or V cubed or something like that you take V and send it to negative V and in fact that’s the equation that’s written here on the right you can always derive these directly from the Loren transformation or you can memorize one of them and remember how to transform it into the other by swapping the object velocities and flipping the sign of the frame velocities I’ll leave it up to you as to what your best possible learning strategy is for this but know that if you memorize one of these you can figure out the other from Context and knowing how to trade the mathematics around now what if the object instead of having its velocity aligned parallel or anti-parallel to the frame velocities is moving in a direction that isn’t solely parallel or anti-parallel to V so you might be tempted to assume that the object velocity in for instance the y direction assuming that the frames are moving only along the X and X Prime axis you might be tempted to assume that the object velocity along the y direction and the Z direction as observed in either frame is the same since in the Loren transformation coordinates Y and Y Prime Z and Z Prime are equal to each other if all the motion is along x and x Prime and you’d be wrong you need to be very careful with these things why well think about it a second object velocity necessarily involves the time derivative of a coordinate is time absolute between two different frames of reference well we should feel pretty confident at this point that the answer is that it does not t does not equal T Prime in special relativity because a Time derivative is involved there’s going to be a dydt and there’s going to be a Dy Prime DT Prime and While y may be equal to Y Prime T is not equal to T Prime so let’s go through this I’m going to consider motion component along the Y AIS the frames are moving entirely along X so V in this is still directed entirely along the X and X Prime axes but I’m going to allow the velocity of the object to develop a component uy or uy prime along the Y and Y Prime axes respectively so let’s look at what the transformation of U Prime to U would be for the case of this component along the Y AIS and Y Prime axis so we know that in the rest frame U subscript Y is just dydt it’s the change in the y-coordinate with respect to to time as observed in the rest frame now it’s true that in the Loren transformation if the motion is entirely along x and x Prime that y does equal y Prime so we can replace Dy with Dy Prime and no harm no foul that’s mathematically allowed but if we’re going to substitute for DT with DT Prime we have to use the full glory of the differential form of the time equation in the Loren transformation and that means replacing DT with the quantity I show here in the denominator of this fraction now of course I can divide the top and the Bottom by DT so that I get a uy prime in the numerator and the denominator gamas don’t cancel out in this case though between the numerator and the denominator because y equals y Prime Y and Y Prime don’t depend on a gamma factor to correct between them and as a result It’s actually an easier derivation I feel than for the case of the object object motion component along the direction of travel of the the frame relative to the rest frame um but it it’s not perhaps quite is uh memorable looking now similarly if we have uy Prime and want uy all we have to do is swap uy Prime and uy in these equations and replace V with minus V and so the corresponding equation that tells us what the velocity component in the moving frame looks like given the velocity component in the rest frame uh will be the one I show here and by the way if there’s a component of motion along Z and Z Prime you can obtain a similar equation it has pretty much exactly the same form as the one shown here with uy replaced by u z and u y Prime replaced by u z Prime um you can very quickly write that equation down but I I just want to go through this because it’s important to recognize that while it’s true that y equal y Prime and Z equals z Prime when the motion is entirely along x and x Prime it is not true that uy is naively equal to uy Prime and that’s because a Time derivative is involved and time does not pass the same way in the two frames when one is moving relative to the other finally let’s take a look at one last special case and that is if the pursuing ship shoots a laser beam at the lead ship so what I’ve done is I’ve replaced the red projectile with a red squiggly line to represent an electromagnetic wave light being fired at the lead ship now the lead ship is still moving at a velocity V Vector with respect to the pursuing ship I’ve put everything along the horizontal axis here but now the velocity of the projectile is C because this is a beam of light and so it will always and forever move at exactly the speed of light so the speed of this projectile is now exactly 2998 * 10 8 m/s as viewed from all frames so if the pursuing ship had fired a weapon like this a laser beam a beam of light well we know that the second postulate of special relativity demands that all observers must see light moving at C regardless of their state of motion so does this velocity addition relationship capture that postulate in all of its full Glory well let’s find out let’s assume that the relative velocity of the lead ship to the pursuing ship is 1 12 C and that the projectile speed as viewed in the rest frame of the pursuing ship is C the speed of light well plugging these numbers in uh we can start from the equation where we have the relative velocity of the two frames and the speed of the projectile in the rest frame and we can get the speed of the projectile as observed in the moving frame so all I’ve done is I’ve replaced u in this equation with c because the projectile is a speed of is a beam of light that’s moving at the speed of light and if you do some algebra you can simplify this equation to C minus V all over the quantity 1 minus V / C and if you do a little bit more algebra you’ll find out that this is just equal to C minus V over the quantity 1 / C * C minus V and if you play with this one step further you find out that this is just equal to C the speed of light so in fact we see that V entirely drops out of this equation once the projectile is a light beam the value of V doesn’t matter at all the relative velocity between these two vessels can be any number and it won’t affect the outcome of the calculation V could have been a half C or negative a half c or8 c or 99999 C basically once U equals c v drops entirely out of the equation and we always recover that U Prime equals c as well the second postulate of special relativity is fully obeyed by this velocity transformation equation so to review in this lecture we have learned how to think about object velocities in different frames of reference and how to go from the coordinates of an object that’s in motion to its velocity in different frames we’ve then used that information to figure out how to properly add velocities of objects to frame velocities in special relativity we’ve looked at a couple of case studies of this and seen that everything seems to comport with the postulates of special relativity which themselves comport with observations of the natural [Music] world in this lecture we will learn the following things we will learn how to Define kinetic energy and momentum while incorporating special relativity we will learn about the nature of mass and the concept of intrinsic mass and we will learn about the relationship between energy momentum and Mass now let’s take a look back at Newton’s Second Law from the perspective of classical physics and in particular have a look at momentum or classical momentum in the context of this discussion so in introductory physics you are introduced to the concept of momentum roughly as follows historically it was observed that there appeared to be a conserved directional quantity associated with motion this quantity which we call Momentum is well defined in the classical domain of physics that is low velocities and large scales by the product of the mass of an object M and the velocity of the object U Vector so we arrive at the definition the so-called classical definition of momentum by taking the product of these two things M * U and that gives us P the momentum or linear momentum of that object now in a closed and isolated system perhaps with a whole bunch of different objects I equal 1 to n it is observed that this quantity overall is conserved that is the sum of all momenta of all objects in a closed and isolated system can be written as a singular number the total momentum and that total momentum remains constant no matter what happens inside that closed and isolated system now when a system is not closed and isolated for instance subject to some net external Force F then the full beauty of Newton’s second law is observed to be obeyed by the system that is that the net force acting on the constituents of the system is just given by the change in momentum of that system divided by the change in time or dpdt so Newton’s Second Law fals ma can actually be Rewritten in terms of momentum Concepts as just f equals dpdt now of course we need to bridge from classical physics to Modern physics and to do that I want you to start thinking a little bit about the laws of physics and their invariance under Transformations from one inertial frame of reference to another recall that one of the postulates of special relativity is that the laws of physics should not depend on what frame of reference you are measuring them in they should be the same for all frames of reference the consequence of that of course is that you can’t tell if you’re in an absolute state of motion but the benefit of that is it preserves the forms of the laws of physics for all observers regardless of whether or not they’re moving so if one subjects the classical momentum concept to consideration moving from one frame of reference to another imagine a second frame of reference observing an object moving at speed U Prime and that second frame of reference S Prime is moving at relative velocity V to the original frame s now imagine that this is all closed and isolated and in the rest frame the velocity of the object is U and in the moving frame it’s U Prime and and the conservation of momentum will hold and so for instance if we take the momentum observed in the rest frame for this object so p = m * U and we use the Galilean transformation from classical physics to move to what we observe in the moving frame we find that of course the moving frame will observe P Prime equals M * U Prime and we can relate the momentum in the moving frame and the momentum in the rest frame using a gilean velocity transformation changing U Prime to U minus V and then Distributing that inside the definition here so when we do that we find out that the momentum observed in the moving frame is related to the difference between the momentum observed in the rest frame and sort of the frame momentum itself M * the velocity of the moving frame now if we then consider changes in momentum in the moving frame with respect to Universal and absolute time so DT prime or DT it doesn’t matter which in the classical view of physics we just wind up taking the time derivative of momentum in the moving frame and if we distribute that time derivative to the two terms on the right hand side above we find out we have dudt and dvdt now since the moving frame is moving at a constant velocity relative to the rest frame dvdt is zero that is the moving frame is not accelerating with respect to the rest frame it’s moving at a constant velocity with respect to the rest frame so the second term is zero and we see that we recover exactly dpdt in the rest frame in other words dpdt in the moving frame is the same as dpdt in the rest frame this is Newton’s second law and so we find that this transformation in classical physics leaves the form of Newton’s Second Law invariant at least under Galilean transformations assuming that’s the correct trans transformation of space and time and velocity now this should all work in domains where the speeds are low compared to that of light but we know that the original definition of momentum was predicated on experiments and observations that were all done in that low velocity large scale regime of investigation that is sort of the human scale of speeds and sizes we also know that that wasn’t quite correct the Loren transformation not the Galilean transformation gives the correct way to define relationships between frames great well let’s just take the classical definition of momentum and apply the Loren transformation the correct transformation between frames so when we do this of course we find that the momentum is equal to mass time velocity and we want to view this in the moving frame where the momentum in the moving frame should be mass times the Velocity in the moving frame what well if we insert into this the relativistic transformation of velocities and special relativity we wind up with this nasty thing over here the mass time uus V over the quantity 1 – UV over c^2 that’s the thing we have to insert that contains the velocity of the object as observed in the rest frame and of course the relative velocity of the two frames all right well fine so let’s then transform this into a statement about differential so if I try to write the differential of of P Prime uh in terms of the differential of P the momentum in the rest frame if I do the calculus on this I wind up with this horrible looking thing here and then of course if I do DP Prime DT Prime which would be the change in momentum with respect to time in the moving frame that’s related to the change in momentum with respect to time in the rest frame by this horribly velocity dependent thing here this is bad why is this bad it’s bad because it totally violates the first postula of special relativity the forms of the laws of physics must be invariant across all inertial reference frames but here we see that one frame has that force is just equal to dpdt but in the other frame that very same law is horribly velocity dependent this is not good and rather than throwing the whole concept of momentum out the window what we should do is stop and ask ourselves did we really Define momentum the conserved quantity associated with degree of motion did we do that assignment correctly in the classical regime of physics did we just get the wrong definition is M * U too naive a definition of momentum given now what we know about space and time and invariance in special relativity now in order to come up with a more appropriate and physically correct definition of momentum that is relativistic momentum there are many many alternative approaches to finding the correct definition of momentum textbooks gloss over this because in many cases the framework for coming up with the exact form of this is not really approachable to students at the level uh of a student taking this course so I had to cherry-pick a methodology to motivate where the definition of relativistic momentum might come from and I prefer the method that comes from my colleague Dar aosta so let’s assume that the problem in the original definition of momentum was that of the definition of time used in the time derivative of space momentum was defined as mass time velocity of object U velocity is the derivative of space with respect to time so perhaps it’s that time definition that’s the flaw in the original definition of momentum after all that definition of time did not regard changes from frame to frame as having any appreciable effect on time DT was not necessarily invariant from frame to frame and in fact could have been the root cause of the problem we saw on the previous slide however there is in fact a Time unit that all observers regardless of their states of relative motion can agree on they can agree it exists and it can be measured the same way in a specific frame every time and that is proper time denoted with the letter to so if two events occur and those events are observed by all observers and all frames of reference all observers agree that proper time will be observed in a frame where the two events happen at the same spatial location that is the definition of the proper time it is the shortest time duration measured in any frame by any method of measuring time durations using two events now it’s always possible to find such a frame if you’re not in the frame where proper time is defined you could always accelerate yourself in such a way until you enter the frame where the regularly occurring events that will be used to define passage of time occur at the same place the time in any other frame is going to be given by the relationship between time in that frame and the proper time so in any other frame the time t for a frame moving at velocity V with respect to the proper time frame is simply given by gamma the gamma Factor associated with the motion of that frame relative to the proper time frame times the proper time duration to now we’re talking about inertial frames of reference moving in relative constant velocities with respect to one another and so as a result of that the gamma factors involved here will not be time dependent they are defined using constant velocities of objects or constant velocities of frames relative to one another or both so consider an object moving at velocity U with respect to the proper time frame that in it of itself that object would be a frame of reference that’s in relative motion to the frame in which proper time can be observed so let’s trade the old time derivative in the definition of momentum that is momentum equals mass time the first derivative of space with respect to time for the derivative with respect to proper time that is momentum will now be defined as mass times the first derivative of space with respect to proper time now we want to convert that into any other frame specifically into the frame where the momentum is being measured which may not be the proper time frame and to do that we just substitute for da with the relationship between it and DT and if you do that you’ll find that you now have math time the first derivative of space with respect to time time a factor of gamma so if this is a better definition of momentum one that preserves the second law from Isaac Newton under Transformations from frame to frame then we should be able to show that and the definition that we get from this exercise using proper time derivative instead of just the plain old time derivative is that the moment um of an object viewed from a reference frame is given by the gamma factor of that object relative to that frame times its mass times its velocity as observed in that frame now again I want to be careful here because the gamma factor that appears here is very specific it has to do with the gamma Factor associated with the velocity of that object viewed in the frame of reference the object itself could be viewed as a reference frame of course but because we’re going to start talking about transforming object velocities into other frames moving at speed V relative to the one where we measured it it’s extremely important to realize that there are suddenly going to become multiple gamma factors in your equations some of those gamma factors will relate to the observation of the object and the passage of time relative to its frame of reference and some of the gamma factors will be related to the ative motion of other frames of reference relative to the one in which you’re defining momentum and if that all seems confusing it is and the only way to get better at this is to practice practice practice so the gamma Factor here I’ve denoted especially with a subscript U to indicate that it is not the velocity of another reference frame V that appears in here but rather the velocity of the object itself U and so this gamma factor is defined as 1 < TK of 1 – U ^2 c^2 that’s what gamma with a subscript U is going to refer to now this redefinition of momentum can be demonstrated with a lot of algebraic pain to leave Newton’s Second Law invariant and in fact this is accepted to be now the correct definition of momentum I leave it to the viewer to go through the exercise sketched out on previous slide to transform

the momentum of an object observed in one frame into another frame moving at velocity V with respect to that first observing frame and show that the form of Newton’s Second Law dpdt remains invariant from frame to frame now any good definition of momentum will hopefully respect the observations of the past that at low velocity the classical definition of momentum seemed to be good enough if special relativity is the more correct General framework for describing space and time then in some appropriate limit in this case low velocity of the object we should be able to recover the classical definition of momentum so let’s give this a try and I’m going to begin by writing the gamma factor for the moving object gamma with a subscript U as a binomial expansion and I’ve used this before in an earlier lecture so hopefully the rhythm of this will begin to look familiar the binomial expansion is very useful for carefully stepbystep exploring what happens when you send a parameter of the theory in this case the velocity of an object relative to that of light closer and closer to one of its limits so we’ll start by writing down gamma subscript U with its traditional definition of 1/ the < TK 1 – u^ c^2 and then we can use the binomial expansion approach to write it instead as a series of terms of increasing powers of the velocity over C so the first term is just one the second term is 12 u^2 over c^2 Etc after that you have terms of order U 4th over C 4th U 6 over C the 6 and so forth those terms matter when U over C is very close to one but when U over C is very close to zero those higher order terms really don’t matter so much compared to the lower order or leading terms in the expansion so now let’s write relativistic momentum using this series expansion of the gamma Factor so I have momentum is equal to gamma subscript U * muu which is now this series expansion * m * U and you’ll notice now that I have an extra U to multiply into the series expansion if I take M * U and distribute it to every term in this series expansion i w up with something that looks like this the leading order term now has a dependence on velocity but the subleading term has a dependence on velocity cubed over c^2 and then the terms after that are Velocity to the 5ifth over C to the 4th or velocity to the 7th over C to the 6th Etc and as U approaches zero that is as the velocity of the object gets much much much much much lower than the speed light essentially as its velocity is sent toward zero any terms that depend on U cubed over c^2 or higher in this expansion are going to vanish they’re going to approach zero much faster than that leading term of mu the leading term will dominate the series expansion as U over C gets very small so I can start from this expanded version of momentum using the binomial expansion and in the limit that the velocity is much much less than the speed speed of light only the first term in the series Will Survive the one that’s largest compared to the others as U over C goes to zero and that’s just m * U we have recovered the classical definition of momentum in the limit of velocities that are small compared to the velocity of light so we can proceed similarly now having had some measure of success with looking at momentum as the quantity the directional quantity of motion thinking about kinetic energy which which is the scalar or directionless quantity associated with motion that can also be conserved so let’s begin to think about kinetic energy in special relativity did we really have the right definition in the old days 1/ 12 mv^ s is that the relativistically correct definition of kinetic energy well we can start by looking at the relationship between external forces changes and states of motion work and kinetic energy when an external Force acts on an object and displaces it over some for instance straight line distance s Vector the action of accelerating this object under the influence of an external Force represents itself a unit of energy being imparted to the object and that energy is known as work work done by an external Force changes the kinetic energy of the object it was in a state of some kinetic energy maybe zero and then a force acted on it and accelerated it and now it’s in a different state of kinetic energy because its velocity has changed that means that the work done by the force has had some action in changing the kinetic energy of the object and according to the work kinetic energy theorem the change in the kinetic energy of an object is directly proportional to the work done by the external Force now the work done by the force on the object displacing it over for instance a linear distance s Vector can be written as the dot product of that external force and that displacement now I’m taking some shortcuts here with the form of the work equation this is for a constant magnitude Force displacing an object over a straight line distance that’s not the general form of the work equation and I will use the general form of the work equation in a moment so let’s assume a constant force acts on an object from the perspective of an observer in frame s and the of course the the form of that force and its relationship to the momentum of that object and the changes in momentum of that object will be given by Newton’s Second Law the force is equal to the change in relativistic momentum with respect to time this is now the correct definition of momentum in that frame and used in any other frame preserves the form of Newton’s Second Law which is FAL ma or FAL dpdt now let’s say the force acts over a small displacement a differential of a path DS vector and at any moment it’s related to the velocity of the object and the time over which the displacement occurs via the fact that the object velocity is the change in the path position divided by the change in time in that frame in other words U Vector is DS Vector DT we can write the differential of work the little bit of work done by that constant force over that little bit of displacement by thinking about the definition of work itself in a more general form that is the little bit of work done in displacing the particle over a little bit of path DS vector by a constant force f is given by the dotproduct of F and DS Vector Now by Newton’s Second Law this has to be equal to the first derivative of the relativistic momentum with respect to time that is what the force should be equal to and again that thing is dotted into DS Vector the little bit of displacement but we can replace DS Vector with its relationship to the instantaneous velocity of a particle under the action of this ex external Force DS is just going to be equal to U DT now to simplify this dot product I’d like to assume that the change in momentum is in the same direction as the force that’s applied on the object so the force is entirely directed in the direction of the displacement or the change in momentum or the change in velocity and as a result of that the dotproduct trivially becomes the product of the magnitudes of the two vectors to find the total work done by the force which is to be related to the total change in kinetic energy if I can find the total work being done by this Force I can absolutely relate that to Delta K the change in kinetic energy and perhaps arrive at the form for the kinetic energy what we’re going to do is we’re going to integrate both sides so by the work kinetic energy theorem the change in the kinetic energy of the object whatever equation that is is given by the work done by the force on the object and that is going to be the integral of this equation here the sum of all the little bits of work should add up to the total work and so that equates to taking the sum of all these little bits here and if I pull out all the con and all of this I’m going to wind up with the mass time the integral of U time the quantity D gamma u u the DTs have canceled out here in this dot product leaving us with just a differential of the gamma U * U well that doesn’t look like a very pleasant integral but there is a way that we can get this into a more pleasing form one that’s more easily solved I’m going to start by rewriting this relationship Delta k = w = m * the integral of the speed * the differential of gamma U * speed to get this into an easier to solve form we’re going to integrate by parts to get a final form for the integral this is using the trick that the for instance integral of UV is equal to UV minus the integral of V du so let’s make some identities between this more general form of the equation and the specific stuff that appears in the integral up here I’m going to identify U as being equal to U that’s straightforward I’m going to identify V as being equal to gamma * U when I do that I can then write U * V which I need here as gamma U * U ^2 and then I need V * du well V * du is just going to be equal to gamma sub U * U * du that’s pretty straightforward try this on your own this will help you dust off your integration by parts but you’ll find that the integral becomes the following the change in kinetic energy is given Now by substituting in using the integration by parts trick as M * gamma U * U ^2 evaluated at the end points of velocity the initial velocity UI and the final velocity UF minus the masstimes the integral of gamma U du again evaluated between the initial and final velocities and if you work through all this you’ll get an equation that looks something like this you have this first term M gamma U * U ^2 plus the second term which looks a bit nastier M c^2 * the < TK of 1 – u^2 over c^2 and we are to evaluate this at the end points of the motion so let’s do ourselves some favors here and assume that the initial speed of the object is zero that means that the initial kinetic energy must also be zero whatever the equation for kinetic energy is that’s got to be true the final speed will just set to be U some final speed U that we achieve and at that point the kinetic energy is K so substituting all this in we find out that the kinetic energy is equal to M * gamma U * U ^ 2 + mc ^ 2 * the inverse of gamma U minus mc^ 2 and rewriting this doing some algebraic gymnastics with the gamma factors and mc^ squ you’ll find that this can be simplified to this lovely little equation here the kinetic energy of a particle is simply given by the quantity of its gamma Factor minus1 * mc^ 2 now I’m going to let you show that last step on your own it’s good practice for the gamma Factor gymnastics that you’ll often have to do in these problems we find that the relativistic kinetic energy is just gamma U minus one all * m c^ 2 m is just the mass of the object C is just the speed of light and Gamma U is its gamma Factor relative to the frame in which the object is being observed you can use the binomial expansion trick once again and I encourage you to try this on your own in the limit that the velocity is much much less than the speed of light and you’ll find that the expression reduces to 12 mu^ 2 the classical definition of kinetic energy these quantities for momentum and kinetic energy have all the right behaviors they don’t look like what they looked like in their assumed classical forms they reduce to their classical forms in the appropriate limit and they leave laws of physics invariant where they can be applied now we’ve looked at momentum and we’ve looked at kinetic energy but what about the total energy of an object in special relativity In classical physics the total energy of an object was just its kinetic energy and if it wasn’t moving it was said to have no energy now that’s not entirely true if that object was being acted on by an external conservative Force it’s possible that that object could have some potential energy associated with it for instance if you raise a ball up in a gravitational field it has some now stored potential energy if you let the ball go it will be released and turned into kinetic energy but for a forcef free situation an object at rest really had no defined energy in classical physics is that still true well we can start by just simply noting that as before the total energy of a body in any system is composed of at least two parts a kinetic part describing the energy associated with its motion and a potential part describing any energy that is stored internally in the system and that could be released by some means now the total energy then is the sum of these two pieces so I will use capital E to denote total energy K to denote kinetic energy and youu to denote potential energy or stored energy we see that kinetic energy and special relativity is the difference of two pieces K is equal to gamma mc^ 2 minus mc^ 2 so if we rearrange the above total energy equation and then plug in this expression for kinetic energy we arrive at an interesting preliminary conclusion so if I take K and solve for that using the above equation I find that K is equal to the total energy minus the stored energy and if I substitute in with this equation I find that K is also equal to gamma u mc^ 2 minus mc^ 2 and by identifying and relating terms in these two equations for K I can draw the conclusion that the total energy of a object is given by gamma u mc^ 2 and the stored energy of an object even one that’s at rest is mc^ squ it’s mass times the speed of light squared so by this identity the total energy of an object in special relativity is given by gamma c^ 2 and in the limit that the object is at rest we see that the total energy becomes not zero but mc^ squar mass time the speed of light squared and we note that the same quantity mc^ 2 has been identified in the above exercise as a kind of energy stored somewhere in the object what’s particularly remarkable about this exercise is that by our own means we arrive at a conclusion that Albert Einstein too arrived at in his miracle year in 1905 it’s one of the most profound conclusions drawn from special relativity that mass is itself a form of stored energy and even when a body is not moving it’s to total energy is not zero but rather decreases to a minimum given by e equal mc² and this latter equation is one of the most famous in the history of science it is an equation that would lead to the development of nuclear weapons nuclear power plants the Pet Scan a non-invasive medical invention the particle collider and many other Technologies taken for granted feared or loved in the modern world for an indivisible fundamental particle for instance the electron is a pretty good example of this we’ve never seen that the electron is made of anything else one has to conclude then when when it’s at rest its energy is the result of some kind of intrinsic Mass a fundamental property of matter just like electric charge appears to be a fundamental property associated with matter now it’s it’s interesting to ask yourself well how much energy if I could find a way to convert it into some other form is contained in the mass of an object well consider the fact that a typical is human being has a mass somewhere in the realm of 60 kg and if by some means all of that could be converted to another form of energy like kinetic energy or chemical energy or radiation then the above equation tells us the energy in Jews that this represents e = m * c^2 which is 60 kg time something that’s about 9 * 10 16 M s/s squar this yields of total energy in Jews stored in your body in the form of mass energy is 5.4 * 10 the 18 jewles now for comparison the energy the little sliver of energy that reaches the Earth from the Sun every second a tiny bit of the total energy that the sun can Emit and yet the same energy that keeps our planet warm and hospitable to Life as We Know It that energy is 10 the 17 jewels the stored energy in the form of mass energy in your body is a factor of 10 more than that and if it even a fraction of it could be converted into some other form of energy it represents a terrifying amount of potential so let’s do an example of this sort of hidden energy of matter by considering the mass that’s lost by a uranium nucleus during fision the process of breaking that nucleus into pieces nuclear fion was itself first discovered by physicists and chemists OT Han and frit stman and this was done in Germany in December 193 38 if you know anything about your history this was the period of Nazi rule of Germany now the observation of nuclear fion especially the fact that the uranium nucleus was observed to split into nearly two equally massed Parts was a bit of a mystery and it was explained very quickly thereafter by physicist Lisa miter and her nephew OT frish the physics Community came to understand that what was going on here just in breaking up the nucleus of a uranium atom was the potential of a vast power that lays in the hearts of all unstable atoms to be Unleashed on humankind so consider the process shown at the left this little blue ball is supposed to represent a neutron one of the components of a nucleus they can be freed from the nucleus and fired at other nuclei a neutron striking a u235 nucleus will set off a chain of events that results sometimes and it breaking up into roughly equal Mass pieces a nucleus of the element Krypton Krypton 92 and a nucleus of the element barium barium 141 now the mass of the unsplit u235 nucleus is given in atomic mass units using this number and I’m keeping the Precision on purpose because small differences when it comes to mass energy matter a lot now the masses of the daughter nuclei Krypton 92 and barium 141 are 83. 798 Atomic units and 137.225 atomic units respectively now I should note that for purposes of conversion one atomic mass unit is given roughly as the mass of a proton 1. 16605 42 * 10us 27 kg now if you check the daughter masses do not add up to the parent mass mass is not conserved in this process it’s lost in the fision process and the amount of mass that is lost is roughly 14 Atomic units even accounting for the fact that three neutrons get produced in the fision process that only adds up to roughly three atomic mass units that’s still 10 atomic mass units or so of energy left over that could be converted into forms like kinetic energy or heat now since we’ve checked that the daughter masses don’t add up to the parent Mass we recognize that there’s missing mass energy here and that mass energy that’s missing is about 2.1 * 10- 9 jewles about a billionth of a jewel now that doesn’t sound like much but consider what’s going on in this cartoon at the left three neutrons have also been produced in this process three neutrons that are bullets that can be fired at other u235 nuclei that might be lingering nearby for instance if you highly enrich uranium to greater than 90% pure u235 it’s possible to set off a reaction of events that cannot be stopped and has catastrophic consequences this process can initiate what is known as a chain reaction as you multiply the fision process over and over and over again using these Neutron bullets that get produced from the first fion process so for instance the first split makes 3 to the 1 neutrons the second generation of splits makes 3 to the 2 neutrons because each of these neutrons goes on to split a uranium nucleus that produces three neutrons so that gives you nine the third generation gives you 3 to 3 or 27 a typical chain reaction in purified u235 can go something like at least 40 to 50 Generations before or this device will blow itself to Pieces that’s a multiplicative factor of about 3 to 45 or 3 * 10 the 21 so you’re taking the energy left over from one split and you’re multiplying it by about 10 to the 21 now those neutrons won’t all go on to split uranium nuclei some of them will be thermalized and will result in dumping thermal energy into the body of the material or into the surrounding air around it if the energy of those neutrons is converted to heat from collisions you’ll find that this level of multiplication is sufficient to explain the explosive yield of the very first uranium Atomic weapon code named little boy which was equivalent to about 13 to 18,000 tons of TR Nitro toine or TNT being dropped on a single City that’s 54 to 75 trillion jewles of energy that weapon devastated the Japanese city of Hiroshima at the end of World War II so we can see that a little bit of mass energy goes a long way and it can have positive applications in society you can have negative applications in society but all of this stems from the Revelation that energy and mass are not distinct from each other now in classical or Newtonian Galilean physics there is a relationship between momentum and kinetic energy we know that k equal p^2 2 M go ahead and try it yourself if you’ve never seen this before convince yourself that this is true in classical physics P = MV K = 12 mv^2 do the substitution there’s a relationship between kinetic energy and momentum now in the more correct description of space and time given by the special theory of relativity we have kinetic energy mass energy and momentum what is the correct relationship between these things let’s begin with the momentum equation that is momentum is equal to to gamma U * m * U let’s then insert a sort of clever multiplicative one multiply this equation by C over C which has the effect of just multiplying the equation by one but allows us to distribute the C in a useful way we can take the denominator 1/ C and move it to the left and Associate it with the velocity of the object U so we wind up with a term of U over C in this equation now we know that the equation for total energy has c^2 and Gamma u in it and Gamma U depends on u s over c^2 they’re related to each other so I recommend you you try squaring this above equation Square P which then squares this thing on the right hand side gamma U * m * U over C * C and when you do that you get this equation here now if you then use the fact that u^2 over c^2 can can be related to gamma by 1 – 1 / gamma ^ 2 you can then insert that and you find that P ^2 is equal to m^2 c^ 2 * the quantity gamma U ^ 2us 1 now if you stare at this for a moment you’ll notice that this equation has a piece in it that’s awkwardly close to e^2 e^2 the total energy would be given by gamma ^ 2 m^2 C 4 so multip both sides of this equation by c^2 we wind up with p^2 c^2 on the left this is going to be equal to m^2 C 4 * the quantity gamma ^ 2 – 1 if we then distribute the m^2 C 4th into the parentheses we wind up with this equation and we can identify the first piece here as e^2 and the second piece here as m^2 C to the 4th or the square of mass energy so putting it all together we find that energy and mass and momentum have a relationship to each other and it’s an elegant relationship between an object’s total energy its momentum and its mass energy in special relativity and that relationship is given by this quadratic equation e^2 = p^2 c^2 + m^2 C forth now this equation allows us to think about some cases of certain kinds of particles and one very interesting special case is to look at particles that have no intrinsic mass now the electron is a particle with intrinsic Mass the muon is another example of a particle with intrinsic Mass albeit 27 time that of the mass of the electron but we can ask ourselves what if there is a part particle out there in nature that has no intrinsic Mass can it exist and if it did exist what would its properties be well let’s take a look at that we can use these relationships to study this very special case now it will turn out that photons which are the particles involved in light have never been observed to have an intrinsic Mass they behave as if they have no mass at all so let’s go ahead and take that exact limiting case of m equals 0 and if we plug that into the energy momentum and mass energy relationship we find that we’re left with e^2 = p^2 c^ that is we can take the square root of this and say that the total energy of a massless particle is given by its momentum times the speed of light the total energy of a massless particle is entirely energy of motion in other words if such a particle could be stopped from moving you would have to interpret it as them ceasing to exist their total energy would suddenly become zero but of course that violates the conservation of energy you can’t just make energy go away without consequence so this implies that such particles can actually only be stopped when they’re removed from the natural world by being absorbed into another process now you might then feel emboldened by this and say aha well this is great uh I’m going to go ahead and figure out what e and P are for massless particles but then you very quickly run into a problem and that is that e depends directly on M and P depends directly on M as defined in special relativity and so you get no useful information from these equations from special relativity special relativity can’t give you otherwise useful information about what the total energy actually comes from and what the momentum actually comes from for such particles so what is it that defines energy and momentum of a common particle like a photon which so far as we know has no intrinsic Mass no mass energy well to answer that question we’re going to have to wait a little bit longer and see as we enter the next phase of this course so to conclude this lecture let’s look at what we have learned we have learned how to Define kinetic energy and momentum while in incorporating the principles of special relativity and in doing so we’ve learned something deep about the nature of mass and we’ve learned to appreciate that there is intrinsic mass in nature and that mass in general is associated with a kind of internal energy of all objects an object at rest does not have zero energy it has internal energy given by MC s we’ve also learned about the relationship between energy momentum and mass we’ve looked at some applications of the relativistic concept of energy momentum and mass and we’ve left ourselves with some questions that we can hopefully resolve by delving deeper into nature in the next phase of the course before we get started on this lecture on the first glimpses into the general theory of relativity I want to kind of put a little cautionary warning label at the beginning of this video in any textbook at the level of a course like this and certainly in this lecture video I don’t want you to walk away with a feeling of full confidence that you have completely ly understood the generalization of the theory of space and time that Einstein set in Motion in 1905 with what we call the special theory of relativity relativity is an extremely rich subject you can quite literally fill volumes on this particular bit of material and in fact I’m holding one of them in my hands now that I’ll show you later the general theory of relativity is fundamentally built on a rich and complex set of mathematics that students at the level of a course like this simply have never seen and cannot be expected to master in a week or a month or even three months without really first having had the full breadth of undergraduate mathematics now that said there are nuggets of of ideas and Mathematics that one can draw out of the general theory of relativity and use to motivate in the context of special relativity the implications of the grander theory of space and time since 1916 when Einstein first established the calculational framework the reliable calculational framework that set the stage for the general theory of relativity in all the work that would be done with it students have struggled with this material because it challenges many preconceived notions many Concepts that we walk into any standard science class cherishing so I want you to be a little bit forewarned first of all that the Nuggets that we will draw out of the general theory of relativity and analyze in the context of special relativity can have some stunning implications that either will challenge things you already believe to be true or which open your eyes to the grander scale of the cosmos that we inhabit now I I mentioned this book earlier this book is one of the seminal works in the field of physics on the whole of the general theory of space and time and it’s entitled gravitation its three co-authors are Charles misner Kip Thorne and John archabald Wheeler now all three of these individuals each in their own way are considered some of the brightest lights of 20th and 21st century theoretical physics and this book is expansive in its treatment of the subject I mean look first of all at how thick this book is and if you flip through this you will quickly see that most of us would be out of our depth in the level of mathematical rigor and notation and variety and subject matter that is minimally expected in order to follow along with a text like this certainly to its Bitter End all of this is to Simply point out that the general theory of relativity is complex and Rich and mathematically far beyond the scope of a course at this level now that said we can draw buets of ideas out of the general theory of relativity and we can put them in context in our own course experience for a course at a level like this one in modern physics now some of the names on here may seem familiar to you Kip Thorne for instance has become recently famous not only for winning the Nobel Prize in physics for one of his key bits of work on space and time energy and matter and the theory that links them together in general relativity but also because he has served as an advisor to film and TV including things like the movie Interstellar from 2014 which had some of the most advanced visualizations of physics based on the general theory of relativity in any movie that came before it or since then John Wheeler is another bright light in the field of theoretical physics he will feature brief briefly later on in this lecture in the context of an individual who could not only deal with the mathematics of this subject but elegantly communicate to an audience even at the level of our course the grandest sweeping summary of the general theory of relativity and its implications for energy matter space and time so with all of those caveats in mind let’s pluck some interesting nuggets of ideas out of the general theory of relativity place them in a local context and special relativity where we feel more comfortable with the mathematics albe it with the caveat that the mathematics required to really do this treatment is far beyond our grasp at this stage in a Physics course at the University level without the full breadth of undergraduate mathematics behind us quite yet let’s see what those nuggets tell us should be revealed about energy matter space and time and then let’s look at how those ideas have implications for the whole structure of the cosmos in which you and I live so with all of those things in mind let’s start digging into some of the basic ideas that motivated the general theory of relativity and take a look at some of those nuggets of ideas that we can couch in the picture of special relativity that we’re a bit more comfortable with at this stage of our engagement in physics in this lecture we will learn the following things we will learn about the transition and thinking from the special to the general theory of relativity this will by no means be comprehensive but merely a taste of some of the basic ideas that led Albert Einstein ultimately to construct this theory of space and time we’ll look at some implications of the general theory the of Relativity on specific physical phenomena and we’ll look at some of the large scale implications for space and time let’s talk a little bit about the transition from special to general relativity experiments on the speed of light in the first and second Decades of the 1900s continued to yield no disconfirming evidence for the postulat of special relativity now Einstein’s physics work in 1904 and 1905 and Beyond earned him the faculty position that he had so richly sought after his PhD and he was able to escape the job as a patent clerk in Burn Switzerland and finally take on the mantle of the academic position that he had hoped for after earning his PhD he was sure that the special Theory itself self could be generalized to a complete theory of space and time including he hoped an explanation of the nature of gravity itself the very prize that had eluded Isaac Newton all that Isaac Newton could establish was that his law of gravitation correctly described the behavior of gravity on all scales that could be observed at the time Einstein ambitiously pursued The elusive prize that Newton could not grasp and that was to finally unmask the nature of gravity itself now this work would take another decade of struggle Einstein would fail many times and in fact if you look at the record of Einstein’s work in the decade that followed his miracle year there were some serious missteps in papers that he had published based on his work beyond the special Theory of Relativity and had anyone been able to mount an experimental test of the claims of the general theory of relativity prior to 1916 Einstein might have been laughed off the scene of physics but it took more than 10 years to be able to make an observational test of one of the key predictions of the general theory of relativity and that ultimately saved Einstein buying him the time he needed needed to fully develop the mathematics behind the general theory put on firm ground theoretically speaking the predictions of the theory of general relativity and to finally publish key papers in 1915 and 1916 that are considered the first accurate and fairly complete treatment of the subject from Einstein himself now interestingly in order to lay the groundwork for the general theory of relativity Einstein required much of the advanced math that he had issued during his time in graduate school one of the things that offended many of the faculty that had him in his classes he was forced then to go and actually relearn subjects that he had actively avoided in some cases during his PhD education he benefited from a close network of friends who were outstanding physicist assistant mathematicians in their own right and through his network of friends he was able to build his own Foundation strong enough to eventually lead to his key insights and firm mathematical grounding of the general theory of space and time in this lecture we will explore some of the very basic ideas and tease the larger implications of the general theory it’s very difficult at this level as I have warned you before to give you the full treatment but I will do my best with the aid of the textbook that we use in class to attempt to communicate some of the key nuggets and frame them in the language of special relativity which we’ve developed more carefully over the last few weeks let’s begin with a tale of two masses we take for granted in introductory physics that Mass appears in a large number of equations but if you really boil it down mass as a concept appears in two distinct equations in introductory physics the rest of the equations that we use that involve Mass can be found to stem from these two laws of nature now what’s interesting and what may not have been pointed out in introductory physics was that the two equations do not necessarily have anything specific to do with one another as regards Mass itself the two equations in question are Newton’s second law of motion FAL ma where Mass appears as the multiplicative scalar factor in front of acceleration the constant of proportionality between what an external force of a general nature exerts on a body and the responding change in the state of motion the acceleration of the body but Mass also appears in an equation that describes the nature of a very specific fundamental Force Gravity the law of gravity states that the force of gravity between two bodies which we might label one and two is proportional to the product of their masses divided by the square of the distance between them now the M for mass that appears in Newton’s second law has to do with inertia the tendency of a body to resist changes in the state of motion and so it’s more honest to say that Newton’s second law is concerned with a mass concept we might label inertial mass the mass of a body that resists changes in state of motion but the m that appears in Newton’s law of gravitation has to do with the primary cause of the gravitational force between two bodies that have mass this doesn’t necessarily have anything to do with their tendency to resist changes in the state of motion it has all to do with the degree of the gravitational attraction between the two bodies this is more honestly referred to as gravitational Mass potentially distinguishing it from inertial Mass there is nothing in these two laws that says that these two quantities these two kinds of Mass have to be fundamentally the same and yet their equivalence the equivalence of inertial mass and gravitational Mass has been tested to a remarkable precision inertial mass and gravitational Mass appear to be one in the same let’s take a look at this by briefly stepping through some mathematics now couched in the language of inertial mass as potentially distinct from gravitational mass and revisit some conclusions we drew in introductory physics so a consequence of their equivalence is often taken for granted what if they weren’t equivalent well if they weren’t equivalent then we would rightly state that two bodies that are acting on one another through the gravitational force can have their degree of acceleration explained by Newton’s second law but without necessarily equating inertial mass and gravitational mass for example if we consider the Earth to be pulling on a body say you you jump off the surface of a table in an attempt to accelerate down to the floor and land on the ground the Earth is attracting you down toward its Center we can figure out the local degree of acceleration due to the gravitational force by taking the product of Newton’s gravitational constant G the mass of the Earth and dividing by the distance between U and the center of the earth squared now we multiply that acceleration which we often denote little G for gravitational acceleration times your gravitational mass and we set that equal to your inertial mass times your total acceleration now if we then solve for the acceleration due to gravity we find that this would be equal to this gravitational acceleration little G or Big G times the mass of the earth divided by the distance between U and the center of the earth squared times the ratio of of the gravitational mass and the inertial mass and if we were to substitute for little G 9.81 m/s squared here we we would conclude that if gravitational mass and inertial mass are not the same if for instance the gravitational Mass were 10% of the inertial mass that only 10% of your inertial mass has anything to do with causing the force of gravity well then we might conclude that your acceleration might be very different than a body that has more mass but we don’t observe that when two objects of different masses fall in a uniform gravitational field a field of uniform gravitational acceleration if you will all bodies even though they possess of different masses appear to fall at the same rate and so by I you can already draw the inference that gravitational and inertial mass are if not equal to each other very close to one another and as far as the limits of our ability to test this have taken us we’ve never seen a difference between gravitational and inertial mass they really seem to be one and the same within the limits of experimental methodology this leads us into one of the key insights that Albert Einstein had early on in the process of trying to generalize the ideas of space and time to include gravity this is summarized by the phrase the equivalence principle now I’ve pointed out that observationally there seems to be an equivalence of gravitational and inertial mass and this can lead you down the path as it did along with some thought experiments for Albert Einstein to a larger consequence and that is the principle of the equivalence of a system accelerated by a constant force or alternatively experiencing a constant gravitational field the principle of equivalence in the language of Relativity and space and time is about the equivalence of two different situations one in which a system is experiencing an external force of some kind that causes it to change its state of motion like a a rocket or something pushing on something else the equivalence of that system and a system that’s experiencing a gravitation acceleration that pins objects to a low point in the system and I will illustrate this Einstein observed early on in his thought process about all of this that due to the equivalence of these two kinds of mass inertial mass and gravitational Mass there’s really then no difference between being under the influence of a uniform and constant gravitational field or source of gravitational acceleration or instead being placed in a non-inertial reference frame one where there is an observed net force acting on all the parts of the system by the action of some other kind of external Force this picture illustrates the idea in a rather cartoonish but elegant way the scenario I like to have that goes along with these set of pictures is the following imagine you wake up and you find yourself in a room with no doors and no windows there’s no way to see past the walls of the room at all as I like to joke with people this is like the premise of the opening scene of some kind of cheap horror movie you push yourself up from the ground you feel gravity pulling you down and you have to work against gravity to raise yourself up that’s what it feels like to you now on the floor next to you was a red ball you leaned down down and you lift up the red ball and you hold it out roughly at arms length and level with your shoulder and you let the ball go and indeed You observe that the ball falls down to the floor of the room you then check your watch think about the average height of a human being measure roughly how long it takes that ball to make that drop from shoulder height to the ground and you’re relieved to find out that you seem to still at least be on earth albeit you have no other external information to tell you where you are because the ball appears to fall at a rate of acceleration consistent with g at the surface of the Earth 9.81 m/s squared but in reality in this opening scene of this cheap horror movie the camera zooms out and gets a view from outside of the enclosure in which you have woken up and reveals that you’re not on Earth but rather far from all planets and stars in empty outer space being accelerated upward from your perspective by a rocket that you can’t hear through the soundproof and vibration prooof walls of your little prison and that rocket is accelerating you Upward at 9.81 m/s squared so from your perspective in the sealed room you think that objects are falling down in a gravitational field or that you have to work against a gravit field to lift yourself off the floor but in reality what’s going on is the entire system is being pushed by an external Force experiencing an acceleration in One Direction of 9.81 m/s squared which gives you the illusion inside the room that you’re in a gravitational field even though you’re not how would you be able to tell the difference between these two situations a soundproof vibration prooof windowless doorless room with no external reference information to tell you that you’re moving or not in outer space and a gravitational field on Earth under the same conditions where yes you’re on the surface of a planet but you have no external information that tells you that a ball dropped in either of those two environments will look and behave the same way and it was this Insight or a variation on it that led Einstein to realize that a constant acceleration due to gravity is no different from taking a reference frame and accelerating it at a constant rate as a visual test of the equivalence principle let’s see if you can tell the difference between the following two situations a zero gravity environment and an environment that’s in freefall in a gravitational field take a look at the video on the left and the video on the right which one do you think is shot in a zero gravity environment which one do you think is shot in a freefall environment where a gravitational field is present the answer is that neither of these is in a zero gravitational field environment this may surprise you maybe you recognized somebody in one of the videos and said aha that person’s an astronaut therefore this video must have been shot in zero gravity but in fact both of these videos are depicting life in a locally inertial reference frame in freef fall in a gravitational field the video on the right is shot in something known as a reduced gravity flight an airplane that makes a parabolic Arc through the sky and briefly enters freef fall in the earth’s gravitational field close to the ground the video on the left is shot in the International Space Station the International Space Station may be far above the surface of the Earth but the acceleration due to gravity is actually quite strong in its orbital position however the International Space Station is orbiting the Earth every 90 minutes and as a result of this circular motion it’s actually in freef Fall constantly it’s just missing the Earth because it’s moving to the side every time it falls down a little bit it’s almost impossible to the human eye to tell the difference between life in a freefall frame of reference in a gravitational field and life in a zero gravity environment that’s no accident that’s the equivalence principle in action Einstein then defined the concept of a locally inertial frame by imagining not this situation I’ve described here but a system in which a person for instance is in freefall in an external gravitational field the concept of a locally inertial frame of reference is one in which all parts of the system are experiencing a constant acceleration due to gravity but because all parts are accelerated the same way it’s as if the system is entirely free of any external forces it’s as if everything is in an inertial reference frame with no external forces because all of you are accelerating at the same rate at the same time this is an incredible Insight it may not seem that impressive but it frees you very suddenly from thinking of gravity and acceleration of an entire reference frame of things as different things and it was this Insight that freed Einstein to think about gravity in a completely new way as another aspect of space and time the key idea here is that without external information in any of the situations I’ve just described being in free fall above the surface of the Earth um or being inside a sealed room with no windows and no doors that’s vibration resistant and sound resistant without external information there’s absolutely no experiment you could do in any of those situations that will tell you that the system is either away from a gravitational Source or simply in freefall in a gravitational field these all seem like inertial reference frames as a result of that now since there is no difference between gravitational acceleration and the ACT of changing a whole reference frame into a non-inertial reference frame you can analyze phenomena in a situation for instance where an inertial frame of reference is considered instantaneously inertial that is although it experiences overall some acceleration like taking a whole room stopping a rocket to the bottom of it and accelerating the entire room and all its contents up at 9.81 m/s squared you’re doing that equally and fairly to all parts of the system and so so at any moment in time all elements of the frame will have the same velocity now of course if you take an object off the wall of this frame of reference hold it out and drop it it will appear to fall down because once you let go of it it’s no longer part of the frame of reference it’s not bound to it in any way and so uh it will appear to fall down as if under the influence of an external gravitational field but of course what’s really happening is that the floor of the reference frame is being accelerated up toward the object now read from the bonds of the reference frame at 9.81 m/s squar perspective is everything it’s the relativity of whether you’re falling in a gravitational field or whether the floor is rushing up toward you at the same rate it’s the ambiguity in those perspectives that lead to the key insights that blossom into the general theory of relativity so in order to help us to picture this let’s consider reference frames in the same way that we’ve done the this in special relativity before let’s imagine a frame that’s we’re going to always take to be exactly and absolutely at rest we choose which frame that is and then we Define it as the rest frame it’s our choice to make it doesn’t matter which one we pick I’m going to choose this one with X and Y coordinates as the absolute rest frame now the frame over here with our friendly observer in it it’s labeled as having its own axes X Prime and Y Prime for instance uh and I’ve exaggerated the x-axis here only because I’m going to need some room on this as I start to play around with it but at first at Time Zero in our little thought experiment here the rest frame and this Frame are in the same state of motion the velocity of what will become the moving frame is instantaneously zero at time zero and so it to instantaneously at Time Zero is a rest frame but this Frame which I’ve labeled with prime notation ation is actually experiencing a net constant acceleration a and in the next instant of time its velocity changes from zero to something non zero in this case it goes to being a Teensy bit above zero a little differential of velocity DV above zero in the IAT or positive X Prime Direction so it was at rest same as the actual rest frame and an in later it is no longer instantaneously at rest instantaneously now it represents a moving frame S Prime at velocity DV relative to what we consider the actual rest frame but that acceleration continues to act and so in the next instant of time the velocity is increased again to twice DV and so now while at this instant of time it’s another inertial reference frame albeit with a different velocity relative to the rest frame it is the result of an acceleration that has been acting on the system the whole time from Time Zero to time one to time two at each instant in time this frame is inertial because it has a well-defined velocity at that moment with respect to the rest frame but overall we can clearly see that this is a non-inertial reference frame one that is experiencing a net acceleration and a person in that frame would conclude overall that there must be some external force acting on the system because they will see objects freed from their frame of reference to behave as if an external force is acting on them now let’s start to dig a little bit into some of the implications of the more General view of space and time now that we freed ourselves and allowed for the equivalence of a gravitational force to a frame that’s non-inertial experiencing an external acceleration by any means necessary Now using this imagery of frames that are instantaneously inertial but overall non-inertial frames of reference let’s analyze an observation of light that has been emitted during this period of slight accelerations of the frame of reference S Prime so let’s consider a light source this black dot here that’s pegged to the Y AIS of frame S Prime it’s fixed in that frame it’s bolted to the wall we can consider the y- AIS to be a wall in the frame of reference the x-axis is like the floor of the frame of reference the person is firmly rooted on the frame of reference and they’re experiencing only a slight acceleration it doesn’t totally knock them off their feet to be accelerated it’s a very gentle acceleration the speeds that we will consider always in these examples will be very much less than the speed of light this will help us to get at the implications of general relativity without having to dig into the full General mathematics of Relativity which is much harder so this is our situation at Time Zero we have our happy Observer they’re looking at this light source on the wall that could emit a pulse of light at any time and in fact at Time Zero we’re going to allow the light source to send out a wave of light so at that moment T equals z it emits a wavefront but at the same time the frame is accelerating it’s been accelerating and instantaneously at time zero its velocity happened to be zero so it was in our rest frame as we’ve defined it but at that moment Time Zero the light source pulses emits a wavefront and that wavefront of course being light is going to travel at a velocity of exactly C from the perspective of any observer in any frame of reference and it will travel from the left to the right from the uh above the origin of the S Prime coordinate system where the light source is pegged toward the Observer over here at some other coordinate along the horizontal axis in frame S Prime so we have a light wave a wavefront traveling at Sea released from its prison in the light Source at Time Zero now at some time later the light wave will cross the gap between the light source and the light Observer and the light Observer will see it but what will the Observer see because in that time that it took the light wave to cross the Gap the frame has changed its state of motion from zero to some velocity DV will the The Observer see the light wave as it was emitted from The Source or will they see something else let’s take stock of the key elements of this question from the picture that I showed you on the previous slide first the light was emitted originally from a source that was considered to be at rest that light was emitted with frequency F at the source F source and wavelength Lambda also defined at the rest frame of the source now the light source and observer remain a fixed distance apart the entire time in this question because they both accelerate together the system is experiencing only gentle accelerations not enough to knock the Observer away from their spot on the xais so they remain planted at their position the light source is bolted to the wall the whole thing is accelerating together at the same rate so their state of motion is changing instantaneously in the same way at every moment of time and so there is no change in the distance call it capital L between the source of the light and the Observer of the light light travels at C 2998 * 108 m/s no matter the frame of reference in which it was observed it was emitted in a frame that was at rest it will be observed in a frame that is moving but no matter the state of motion or the change of the state of motion of that frame of reference The Observer if they were measuring the speed of this wave is always going to say it moved at C the light will take a finite nonzero time to travel to the Observer from its source it has to cross a gap that’s going to take some time and by the time the Light reaches the Observer they will have entered a state of nonzero velocity from the cartoon on the previous page they went from zero to 0er plus a little bit due to their acceleration and I’ve called that little bit a differential of velocity DV therefore the light wave in the end will be observed in a frame of reference that is now moving with respect to the frame of the source which had been a rest frame what does this sound like this sounds an awful lot like a Doppler shift problem light being viewed in a frame that’s moving with respect to the original frame of emission this basic Insight then guides the math within the framework of special relativity that we can do to calculate just what the Observer will see so let’s do some very basic calculations with this building on top of all of the stuff that we’ve been looking at over the previous lectures and time in our course I have emphasized this before I’m going to codify it now I want us to assume that we are in an instantaneously inertial reference frame that is at any moment in time we have a definite velocity that’s well defined albe it changing from moment to moment to moment to moment we want the velocity of that frame to be greatly less than c not even close to the speed of light less than 1% the speed of light or even smaller and that’s so that we can have V over C which we’ve previously defined as this nice number beta to be much much less than one this is going to come in handy very quickly in this problem let’s assume again as I pointed out before that the distance from the light source to The Observer which is fixed this whole time is some length L because the light source travels in the same frame as the Observer L remains constant the whole time the light will take a time which will denote delta T T2 minus T1 T1 being the time of emission T2 being the time of observation and that’s going to be given by L / C light has to travel across a gap of length L it does so at a fixed speed C the speed of light the time that will take is L over C full stop now in that time delta T the frame of the Observer will have accelerated by an amount a up from rest to a velocity V and we can actually then analyze this using the very same equations of motion from introductory physics which are still valid here for the conditions that we’re we’re looking at there we could relate initial velocity to final velocity by considering the acceleration of a system and the time over which the acceleration acts this equation will do nicely the final velocity V will be equal to the initial velocity v subscript 0 or V not plus a term that’s the acceleration times the time that has passed over which the acceleration has acted now we can plug in some specifics here for us V will be equal to zero the initial velocity of our instantaneously inertial reference frame plus the acceleration time L over C which is delta T the time it takes for the light wave to get from the source to The Observer this then leads us to the conclusion that V is equal to a * L / C and if we transform this into a expression for beta we find that beta which is V / C is given by the acceleration times the distance divided by the speed of light squared now let’s take this information and let’s put it into the context of the Doppler shift the relativistic Doppler shift specifically the special relativistic Doppler shift so we’re going to treat the case of small velocities relative to light beta is a small number the Doppler shift of the light wave by the time the Observer sees it will be given simply by what we did before we take the frequency of the source we multiply it because the Observer is is in a frame that becomes a frame that’s moving away from where the source was we have to multiply by the square root of the quantity 1 minus beta / 1 plus beta this represents a lengthening ultimately of the wavelength of the light a red shift but we want to get acceleration of the frame the distance between the light source and the Observer and the speed of light into this equation we want to put these things from our picture into this equation and the way we can do that is by doing some binomial expansions of the numerator the square < TK of 1 minus beta and the denominator 1 over the < TK of 1 + beta well if you do those two things and multiply them do the binomial expansion of the < TK of 1 minus beta of 1 over theare < TK of 1 + beta multiply those together you find you get expansion products that look like this 1 minus a half beta plus terms that are higher order in in beta and 1 minus a half beta the same thing again plus again higher order term terms in beta which I’ve just left out but indicated that they’re supposed to be there from these three dots the product of these things is multiplied by the frequency at the source of emission now because we’re working in the case that beta is a number much much much much much smaller than one because V is much much much much much smaller than C we only have to keep the leading terms in all of this and if we multiply out this product and then only keep the the the leading terms in in beta we wind up finding out that this product is approximately equal to just 1 minus beta all times the frequency at the source of emission now we have an expression for beta in terms of the acceleration of the frame the distance between the source and the Observer and the speed of light and if we plug that in we get a final form of this approximate equation for the frequency that the Observer should see the observer in the moment after the light has been trans Ed they accelerate they get up to a a velocity V relative to where the source had been and then they observe the light they will see the frequency shifted by an amount of 1 minus the quantity Al over c^2 this represents a shortening of this represents a decrease in the frequency relative to the source or an increase in the wavelength of the light you can play around with this yourself and convince yourself that that’s the case but we basically conclude that the Observer who now at the moment of observation has been put into a new frame that’s not that in which the light source was at rest when that emission had originally occurred uh will now observe the light to appear shifted from its source frequency and in this case it’s a red shift if the Observer winner frame accelerating in the opposite direction in the direction from The Observer toward the source rather than from the direction of the source toward the Observer so a becomes minus a then the light would instead appear blue shifted shifted to smaller frequencies or shorter wavelengths so let’s think about light as viewed in an accelerating frame of reference we found by making this approximation that we have a frame of reference that’s all accelerated at once so that the light that was emitted at Time Zero is a observed by an observer in a frame that’s no longer at rest with respect to where the source of admission had been that the Observer will see a frequency as they accelerate to the right in the direction from The Source toward the Observer they’ll observe A reduced frequency of the light a lengthening of the wavelength but let’s dig back to the equivalence principle the equivalence principle states that there is no difference between an entire High frame of reference that’s all experiencing an acceleration due to some external force and a frame of reference that is merely experiencing an external gravitational acceleration as a result of the equivalence of these two things one is forced to conclude that the shifting of light must also occur in a gravitational field of acceleration in other words if the source of that acceleration is GRA gravity you know for instance a equal G * the mass of the Earth ID the radius of the Earth squ because you’re

standing on the surface of the Earth that’s just 9.81 m/s Square the old little G from introductory physics and imagine instead we’re viewing light from a source above us we are Downstream in the gravitational field and there is a light source down on the ground below us sort of Upstream in the gravitational field we would conclude that the light as we observe it emitted up from the ground toward our eye must be shifted in this case a red shift in frequency this phenomenon is real it has been confirmed repeatedly by experiments over and over and over again and we’ll look at some of those through problem solving in the class it’s a real phenomenon it must be taken into account when you are dealing with electromagnetic radiation and gravitational fields and it’s known as the gravitational red shift or depending how the problem is set up gravitational blue shift of light in this example if we were laying on the ground looking up at a source that’s above us and looking at light that’s emitted down toward us because we are further Upstream in the gravitational field of acceleration we would see the light in that case as blue shifted it’s equivalent to switching around the acceleration sign now this very same phenomenon the red shifting or blue shifting of light merely because of its transmission in a field of gravitational acceleration has other implications including for the very nature of the passage of time in different parts of a gravitational field of acceleration so by implication from this previous example the Doppler shifting of Light by a gravitational field one can also predict that time itself will pass at different rates at different heights different locations in a uniform gravitational field we saw that the frequency of light in a gravitational field is altered depending on the degree of acceleration if I increase the acceleration of a frame of reference or equivalently increase the amount of gravitational acceleration a system experiences I will increase the Doppler shifting effect frequency of course looking back at the discussion of waves and the Doppler shift and other things related to waves frequency is a measure of the rate at which events happen the time between events effectively so consider observing time at a height zero above the surface of the Earth we’ll call that person the lower Observer somebody right at ground level looking at time passing Say by looking at pulses of light or ticks of a clock or something like that and instead a person who’s way higher up more uh Upstream in a gravitational field of acceleration a higher Observer also looking at their clock or their light pulsing or ticking away from our exploration of frequency and period we know that the frequency of a wave is given simply by one over the period of the wave you can think of that as the passage of time between regular spaced events so the period is just a difference in time it’s a delta T and so really frequency is another way of saying that we’re looking at one over a time difference between regularly spaced events in other words frequency is really probing time structure now if we were to be looking at the time between regular events at our higher altitude in the gravitational field uh this would be related to one over the frequency of events at that higher altitude and we already know how to relate those through the Doppler shift to the frequency of events at the lower altitude we just have to take this Doppler shifting uh equation again do the binomial expansion and we find out that we are just uh multiplying the time duration at the lower altitude by a quantity 1 minus beta and because we are exper exping a gravitational acceleration here height H above the surface of the Earth this is equivalent to 1 – G c^2 in this approximation 1 minus say 9.81 m/s squar times your height above the surface of the earth divided by the speed of light squared you take that quantity and you multiply it by the duration of time between regularly spaced events at lower altitude and you get the time at higher altitude and so as a consequence of this we expect time to pass more slowly for observers who are lower down in a gravitational field if you were to take this to some extreme imagine a person deep down in a gravitational field they might experience an hour but a person higher up in the gravitational field might observe that days weeks or months pass depending on the degree of difference of location in the gravitational field time that passes higher up in a gravitational field is always multiplied by a number whose value is less than or equal to one meaning that less time passes lower down in the field this is a real effect and this effect has been confirmed experimentally over and over and over again and it plays a major role in the operation of Key Systems to Modern existence such as the global positioning satellite or GPS system all modern navigation typically relies on a system of about 24 satellites each satellite orbits the earth twice per day so it’s moving very fast around the earth as a result of that these are not so-called geosynchronous or geostationary satellites that always sit above the same point on the surface of the Earth rather the GPS satellites orbit and they make about two rotations around the earth per day three satellites at any time are required to make a triangulation measurement on the surface of the Earth and they do this using very precise clocks that they carry along with them that have been synchronized to clocks on the ground and this system allows you to make position measurements on the surface of the Earth but the problem is first of all that those satellites are traveling actually very fast relative to the surface of the Earth so they experience a special relativistic time dilation observers on the ground would claim that their clocks are running a bit more slowly than an equivalent clock on the ground because they’re moving and people on the ground argue that they’re at rest so there’s AAL paral relativistic time dilation effect but in addition because humans who are down on the ground making these observations are lower in a gravitational field an observer on the GPS satellite would argue that well okay that’s true there’s a special relativistic effect but there’s also a gravitational effect a general relativistic effect because the clocks on the earth that we’re supposed to be synchronized to are lower down in a gravitational field than the clocks in orbit around the Earth and so for those clocks there’s a general gravitational slowing of time and these two factors must be taken into account in the modern GPS system and in fact any guideline document that you look at for engineering systems for the GPS network will warn you about these Corrections spell them out for you and tell you how to do them so that you can properly synchronize clocks taking into account all of these time effects between the ground and in orbit around the earth these are real effects with real consequences on things like basic day-to-day navigation and without the general and special theory of relativity we would never have understood these had we launched a GPS system before understanding space and time at this level we would have failed to construct a working GPS system now one other implication of general relativity and this can relatively kind of quickly looked at in a cartoonish Way by referring to our uh accelerated frame of reference our sealed vessel um this other effect that we’ll take a look at here is the deflection of Light by a gravitational field now this might seem novel to you but in reality the deflection of Light by a gravitational field the falling of light near the surface of the Earth was not a new idea in the time of Albert Einstein it was actually quickly realized within certainly decades or a century after the work of Isaac Newton had established the laws of mechanics and gravitation that since all objects regardless of their Mass fall at the same rate in a uniform gravitational field think of dropping a wated up balll of paper and a bowling ball at the same time from a few feet above the ground if you drop them so that their bottoms are starting at the same height they’ll hit the ground at the same time the mass of the paper and the mass of the bowling ball seem to play no role in the rate at which their velocity changes as they head toward the surface of the Earth well if Mass doesn’t matter for gravitational acceleration then even one might argue a massless phenomenon like light should fall in a gravitational field now the specific reason why this would happen was put on much firmer footing thanks to the equivalence principle and I’ll walk through an example of that argument here so consider the cartoon at the right we have our sealed vessel it’s sound and vibration proof no windows no doors no way of knowing whether you’re on Earth or far out in space away from all planets and stars now in reality this system is being accelerated upward by a rocket you can neither hear nor feel nor see and it’s doing so at 9.81 m/s squared constantly so you’re in this sealed room there’s a light source on one wall and and you can push a button and fire a wfront a pulse of light across the room so that it strikes the wall on the other side now at the moment that the pulse is emitted and that’s Illustrated here on the left the line connecting its location of emission points straight across the room to a point on the other side of the wall but by the time that the wave reaches the other wall and that’s Illustrated here on the right the wave freed from its connection to this frame of reference that in the meantime has changed its state of motion the light wave will travel on that absolute straight line but from the perspective of a person inside the vessel looking at where the light wave strikes the wall if they had very precise equipment or if the speed of light were much slower than an actually is then they would actually observe that the light wave strikes the wall at a point that’s lower than where it was emitted from so in an external frame of reference that light traveled on a real straight line but the frame moved up in the time during which it crossed the room from a perspective of an observer inside the frame who doesn’t know that any of this is going on they see the light wave strike the wall at a lower Point some vertical displacement below where it was expected to strike that is at the level of the emission source so the light wave reaches the wall but it does so in this case at a lower point now by the equivalence principle there is absolutely no difference between this frame of reference um being accelerated by a rocket or a similar sealed room that’s sitting on a planet experiencing a gravitational acceleration downward of 9.81 m/s and so because of the equivalent of an accelerated frame of reference and a frame that’s merely experiencing a gravitational acceleration light must also fall in a gravitational field because there’s no distinction between these two cases it turns out that this is actually generalizable to any body with mass bending the path of light and this is actually the key Insight that Albert Einstein’s general theory of space and time the general theory of relativity had that helped to distinguish it from Isaac Newton’s original theory of acceleration and gravitation in Newton’s Theory the deflection of say Starlight around a massive body like another star is smaller than the deflection predicted in general relativity which is supposed to be the more correct description of space and time and the way that energy and matter respond to space and time so in the general theory of relativity the degree of flection of light around a massive object by Falling in a gravitational field if you will is twice as big as predicted in Newton’s original mechanical Theory combined with his law of gravitation that’s a key distinction between the two ideas the general theory of relativity and the old theory of mechanics married to the law of gravitation it was that prediction that was tested in the late 19s and led to the confirmation that Einstein’s work was probably the correct description of space and time and energy and matter and this catapulted Einstein into Global Fame it also led to a host of other predictions for other interesting phenomena because light can be deflected by large masses we could imagine being able to see objects that shouldn’t be visible to us using larger rays of telescopes and looking out into the distant Sky we can look for cases where we see an a background object whose light has been bent around a foreground object allowing it to reach our telescope this so-called gravitational lensing allows astronomers not only to see objects that would otherwise be obscured behind other foreground objects things that that sit between us and the thing we want to look at but because the general theory of relativity gives very specific relationships between the amount of mass and the degree of the deflection of light one can use the deflection of light itself to measure the mass of objects with which you can never hope to have physical contact gravitational lensing is one of the many tools that general relativity gives to us as human beings to better understand the universe even parts of the universe that are very old very distant or both so as you can see the general theory of relativity has some fairly impressive large scale implications if you remember something back from your Calculus the second derivative of something with respect to something else tells you about the curvature of the system that you’re studying with the derivative now we’ve considered the fact that space and time are really part of a singular structure they really should be thought of as part of one four-dimensional framework which is called SpaceTime in special relativity we see that space we see that space measurements in one frame can turn into time measurements in another space and time are constantly getting traded for one another or Tangled Up in one another in calculations of motion from one inertial frame of reference to another there’s a link between space and time and that link comes from the fact that they’re really part of one interchangeable four-dimensional framework SpaceTime and it’s in this framework that matter and energy can be described to move and change so general relativity is really a theory of SpaceTime a general broad theory of space and time and ultimately it concludes that what we call the force of gravity is really due to the fact that mass and energy cause space and time to curve or in more colloquial language Bend or warp the second derivative is a sign of curvature and so it should have been a clue that since there’s no distinction between accelerating a frame of reference or subjecting that same frame of reference to an external gravitational field there must be no difference between curvature and gravity and in fact that’s one of the broad conclusions of the general theory of relativity energy and matter curve space and time and so other bits of matter or even light that travel past that object that’s bending SpaceTime will follow the curvature of SpaceTime and the result of this is that from our perspect perspective in three dimensions they appear to accelerate what is a ball doing when you hold it out at shoulder height and drop it it’s not being pulled down by the mass of the Earth rather it’s following a path in SpaceTime that’s curved due to the presence of the mass energy of the earth bending that space and time that is what gravity is that is what Einstein was able to achieve the very thing that Isaac Newton could not grasp the nature of gravity curvature of space and time space and time tell energy and matter how to move energy and matter tell Space in Time how to bend or curve or warp this elegant summary paraphrased from its author is a beautiful way of remembering the implications of the general theory of relativity writ large and it comes from the mind of luminary theoretical physicist John archabald Wheeler the universe is observed to expand in all directions at once and the more distant an object you view in the universe the faster it appears to be moving away from us this tells us that overall SpaceTime is curved now on the grandest scales the largest distances that we can reasonably observe in the universe the universe’s space itself appears to be very flat and smooth but just because space is flat and smooth overall doesn’t mean that SpaceTime is and the expansion of the universe is evidence that SpaceTime itself is curved the curvature of SpaceTime leads us to conclusions about the origin and the fate of the entirety of the universe and it tells us that that the universe as we know it now space and time and energy and matter was born 13.78 billion years ago in an event we have yet to fully understand but which is described by the phrase the Big Bang let’s review what we have learned in this lecture we’ve looked at the transition in thinking from the special to the general theory of relativity we’ve looked at some implications of the general theory of relativity on physical phenomena specifically we’ve considered what it means for light to travel in a gravitational field from a higher to a lower vantage point in that field we’ve concluded that light should Doppler shift either red shift or blue shift depending on the direction in the field That You observe it we’ve also concluded that light should be bent in its path of travel in a gravitational field and we’ve drawn all of these conclusions by using the equivalence principle to map behavior in an accelerated frame of reference onto a frame that’s experiencing an external gravitational acceleration we’ve then looked at some of the large scale implications for space and time the bending of distant Starlight around massive objects that intervene between us in the universe the use of the warping of space and time and the bending of light to infer the mass of objects that we can never hope to weigh by putting them on a scale and the overall implications for the nature of space time as a framework in which energy and matter play out the fact that energy and matter tell space and time how to curve and the curvature of space and time tells energy and matter how to move and how the overall curvature of SpaceTime indicates to us the origin and possible fate of the entire universe itself these Grand themes all stem from the elegant thing thinking of a brilliant physicist who accepted observational evidence from experiment about the nature of light thought deeply about the world around him learned the math necessary to describe the universe and in that elegant language spoke a volume about the cosmos that we are still reeling from today in this lecture we will learn the following things we’ll learn about the concept of temperature of a material body we’ll learn how to establish a scale and measure of temperature about the response of material bodies to changes in temperature and finally about heat energy as the underlying agent connected to changes in temperature there are many things that are left unsaid in the first two semesters of introductory physics we’re only able to cover a prescribed range of topics and that range can be described as follows motion Force the laws of motion relating force and acceleration to changes in state of motion energy momentum the conservation of energy and momentum non-conservative forces oscillatory motion and rotational motion that’s typically what we get covered in the first course in physics in the second course in physics we’re able to cover electric charge electric force electric Fields electric potential and electric currents and the combination of all those things into electric circuits and then we explore magnetic field and force and the basic behaviors of light such as geometric optics or interference and defraction now as a result of this in introductory physics there is essentially no time to discuss the laws of heat energy also known as thermodynamics but nonetheless thermodynamics is an essential Foundation of modern physics It ultimately was a branch of physics that helped to lead the way to Quantum Mechanics the theory of the very small and that is the next subject of this course so in this part of the course we will establish the second half of the foundations of modern physics the concept of temperature the concept of heat energy and some of the behaviors of heat energy we all have a fairly solid familiarity with the various Concepts associated with thermodynamics if you go outside on a day when it’s cold you feel like something is being pulled from your body on mass as if the world around you is hungry to take something away from you and keep it for itself and this feeling this sensation of the loss of something from our bodies where we have to trap it to keep it in is often what we call cold or the concept of a cold temperature of course the flip side of cold is hot there are environments there are situations where instead we feel like something is being put into our body and we want to get rid of it we might shed some clothing in order to help achieve this to help regulate our own sensation of temperature when the world around us is hotter than us we feel that P penetrating into our skin in a way that can be uncomfortable it causes us to sweat and so forth as a mechanism to try to maintain our own uh state of body temperature so cold and hot these ideas are familiar to us even if we cannot articulate the physical reasons why these situations exist now connected to these two things is also the concept of establishing a numerical measure of the degree of hotness or coldness of an environment so for example the average human being and this can vary by age and gender and a number of other factors is typically comfortable especially for intellectual work office work something like that in a temperature range between 70 to 75° fit now a human being experiencing an environment ment where the temperature is observed to be less than that number will often express a feeling of being cold chilly chilled needing to bundle up more to maintain their body warmth on the other hand a person who’s subjected to an environment above that range maybe 85° Fahrenheit instead of 74 Dees Fahrenheit will complain about sweating too much feeling too hot wanting to cool off in some way maybe by by drinking an iced beverage of some kind or maybe taking off a jacket if you’re in a work environment something like that we have a a concept of being able to measure the degree of heat or cold in the world around us including the heat of our own body taking our temperature to see if we have a fever is another concept that is pretty familiar in the human world now connected to these Sensations these experiences we have to come up with a series of critical issues and a plan in order for us to be able to quantitatively describe these scenarios of hot or cold we have a conception of hot and cold we have a conception of that we can measure these things somehow but we need to establish the basis for actually having that quantitative measure that quantitative description of these Concepts how do you know something is hot how do you know something is cold how do you measure that and how do you allow other people independently to establish the same scale of measure let us Begin by establishing that scale on which we can quantify those ideas like a room is too hot or a room is too cold let us then look at the origins of hot and cold and how the underlying concept is really tied to a fundamental concept called heat energy we will close with a relationship between heat energy of a body and its ability to radiate energy away but in this particular lecture we’re going to focus on temperature heat energy and the effects of heat energy not only on the temperature of a body but the structure of a material body let’s begin by establishing a measure of hot and cold now consider the world around you there are some phenomena in nature that appear to occur at very specific so-called thermal conditions that is to say if you could reproduce the environmental conditions under which a particular phenomenon occurs that phenomenon would occur repeatably reproducibly reliably so for example the freezing or boiling in of a body of water the only substance on Earth that can exist in solid liquid and gaseous States under Earth conditions is water it’s essential to Life as We Know It And because it’s able to coexist under a very narrow range of conditions as either a liquid a solid or a gas it makes an attractive phenomenon on which to establish a range of of behaviors that can be used to delineate a scale of temperature measure now that said of course there are materials other than water and they also change in response to temperature for example in the opening lecture video for this series I showed you the result of heating a Bim metallic strip now I’ll return to heating or cooling Metals later but we’ve observed already that two metals bonded into contact with each other will bend curve when exposed to a heat source and that’s [Music] because in this lecture we will learn the following things we’ll learn about the connection between temperature and the constituents of a material body we’ll learn about the precise nature and cause of heat energy and finally we’ll learn about the radiation of energy from a material body now matter is ultimately made from building blocks for example a liquid may be made of a large number of atoms or molecules the atomic theory would not really be accepted as a reliable description of nature until about 1905 but once one adapts the atomic theory as the correct description of material bodies one is then forced to conclude that the large scale macroscopic properties of a material object are somehow connected to the microscopic behaviors of the building blocks from which that material is constructed now the number of things that are used to construct a material body in the human world the max microscopic world is vast for example there’s the concept of the mole one mole is the number of atoms in a 12 G sample of carbon 12 now experimentally you can work it out and you’ll find that one mole’s worth of things anything at all grains of sand planets Stars atoms anything is given by a special number known as avagadro’s number and that number is 6 .02 * 10 23 of anything per unit mole one mole therefore is 6.02 * 10 23 things heat energy must have a connection to the behavior of the building blocks of matter after all if one is depositing a form of energy into a material body that energy must go somewhere and we must look to the constituents of the material body to figure out where energy might be going this helps answer the questions what is heat energy and where does it heat energy go exactly or where does it come from let’s look at an ideal gas as a laboratory for the connection of macroscopic Concepts such as the volume of a material or the temperature of a material and the pressure exerted by a material on its environment to microscopic IC Concepts like the position and velocity of an atom or molecule now we’re going to focus on ideal gases I’m going to start with a very simple simulation of an ideal gas this simulation is provided by the fet demonstration toolkit that’s available on the web and this is a simulator of an ideal gas system now to start I’m going to put one heavy particle of a gas just one atom or molecule of an ideal gas into the system where do the properties of gases like pressure and temperature come from well pressure is force per unit area and so the pressure exerted by an ideal gas on its container and in this case the container is represented by this box outlined here that pressure comes from the force of the Collision of the ideal gas particle with the walls of the container so for example we’ve injected one massive gas particle into the system and we see that it’s bouncing around the inside of the container it collides with the walls of the container and because this is an idealized system we treat it as having perfectly elastic collisions with the walls and the walls do not move and because of this this forces the momentum of the particle the component that’s perpendicular to the wall it collides with to reverse upon Collision so for instance the particle strikes the bottom wall and we see that its vertical component reverses it strikes the right wall and Its Right Moving component reverses to the left we see also that because momentum is conserved in this closed and isolated system that the total speed of the particle remains fixed even if its direction changes and that momentum Chang changes are conserved independently in every direction a collision with a wall to the left or to the right does not change the speed component that is vertical or parallel to that wall so the origin of the pressure of the gas is the force it exerts due to its momentum change on the walls of the container I could now instead inject more particles into the system so now let’s start by injecting 50 gas particles into this system we’ll give them a moment to spread out in the container and we see that while they all come in together as a clump because they didn’t all quite have the same velocity they start not only colliding with the walls of the container but with each other now an ideal gas will have elastic collisions with the walls of the container and with itself and we see that very quickly the gas particles have spread out fairly uniformly throughout the container and they continue to collide collisions will exchange momentum between colliding particles but on average we can see here that the particles are all moving with about the same speed some are moving a little faster some are moving a little slower but collisions level that out and we stare at this for a moment and see that these gas particles all appear to have some average amount of speed and a distribution of velocities that’s sort of spread around that average so slow moving particles can get struck and become fast moving particles fast moving particles can get struck and become slow moving particles but on average it seems like there’s a pretty typical consistent average speed on average we don’t see these particles getting much faster or much slower as a group while an ideal gas is truly an idealization there are many gases that are nearly ideal in nature for instance all of the noble gases es for example helium or argon they behave very much like ideal gases under many common conditions there are even many other substances that under a range of conditions can behave according to the idea of an ideal gas careful experimentation on systems that behave in this sort of Ideal manner have revealed that there is an empirical law that relates the macroscopic properties of a gas the number of moles of gas constituents given by the lowercase letter N the volume of the gas given by the capital letter V the temperature of the gas given by the capital letter T and the pressure exerted by that gas on its containing volume for instance the walls of the container that hold it and that’s denoted by the letter capital P this equation is known as the ideal gas law most students learn this in a chemistry course in either high school or college PV equals nrt the product of the pressure exerted by a gas and the volume of that gas is equal to the number of moles of that gas times a constant times the temperature of the gas now here this constant is denoted capital r it is known as the ideal gas constant and its value is 8.314 Jew per Kelvin per mole it’s named in honor of the French chemist enri Reno therefore the letter R but since a gas is made from small constituents albeit a very large number of them can we connect the microscopic properties of those constituents their positions in Space the changes in those positions in space with time can we connect those to this macroscopic statement about the aggregate behavior of the gas to connect the microscopic to the macroscopic let’s begin by doing what physicists and Chemists in the 1800s did and turned to classical physics after all Newton’s Laws of Motion were the only things that they knew to be reliable as describing nature so why wouldn’t you turn to the thing that had been working for a couple of hundred years already let’s begin with the concept of mass in this ideal gas let’s define the mle mass of a gas as capital M this is simply the mass for every mole of this gas it’s given by adding up avagadro’s number of individual constituent masses which will denote as Little M so if each atom or molecule that makes up an ideal gas each has an identical Mass Little M then the molar mass is that little M time avagadro’s number that gives us the mass per mole of this gas gas what about the volume of the gas well to keep things simple let’s consider a nice cubical space containing our ideal gas it has a fixed size it has sides all of length L and that means that the area of any side of the cubical space the Box in which we’re holding the gas is given by capital A equals the square of the length of any side and that also means that the volume is determined capital V by the cube of the length of any side now pressure is a bit more difficult pressure is the sum total of the force F total per unit area exerted by all gas constituents on the walls of the container at any moment in time an individual gas molecule will occasionally collide with a wall of the volume containing it that Collision will briefly exert a force that force on that area is the pressure now of course a gas is made from many constituents and so it’s the sum total of the average number of collisions per some unit of time that cause the pressure on the walls of a vessel how might we describe this using concepts of motion Newton’s laws and conservation laws all from classical physics well let’s begin by thinking about a single constituent each constituent has a velocity Vector at any moment in time with three components an X component a y component and a z component now since we’re considering an ideal gas we’re talking about elastic collisions between a constituent of mass m and for instance the wall of the container along the x-axis let’s only focus for now on the component of the motion of a gas molecule along the X AIS now during these collisions the wall doesn’t move and so its velocity before and after the Collision is zero and if you consider a single Collision along the x-axis between a gas molecule and the wall that it strikes and if you conserve kinetic energy and momentum as would be true of an elastic Collision then you find that the initial momentum of the gas molecule must be given by its mass time its original velocity in the X Direction and after the Collision conserving momentum and kinetic energy you’re forced to conclude that it has the same speed along the xaxis but it’s reversed the direction of its motion so the final momentum just after the collision with the wall will be negative M times its speed along the xais now a collision results in a change in momentum for the gas molecule and a change in momentum is what is known as an Impulse in introductory physics the impulse is just the difference between the final momentum and the initial momentum and in this case if you crunch the numbers you find out that if we knew the mass of a gas molecule or atom and we knew its velocity just before the Collision along the x axis that the impulse that results from this change in momentum is -2 M VX now if we knew the time over which the impulse occurs then we might compute the force that’s exerted by just this one constituent on the wall and we can do that by relating impulse time and force using Newton’s second law that the force is equal to the change in momentum divided by the change in time what is the time between collisions in one dimension well with a specific wall the time between collisions in one dimension is just the time between when the constituent strikes the wall for the first time bounces back horizontally across the Box strikes the opposite wall then bounces back along the x-axis to the First wall the one on whom we’re considering the force it’s that time the time between the collision with the wall striking the opposite wall and returning to the First wall the time between collisions will be given simply as twice the length of a single wall along the x- axis divided by the speed along the x-axis of that constituent 2 L over VX now the force of the gas constituent acting on the wall will be equal in magnitude but opposite in direction to the force that the wall exerts on the constituent the pressure is the force that the gas exerts on the container what we’ve computed is the force that the gas molecule has experienced by being acted on by the wall we can use Newton’s third law to relate what we have to what we want we want the force exerted on the wall by this constituent we have the force exerted on the constituent by the wall and they’re related in Newton’s third law by a minus sign and if we plug in the force that the constituent experiences because of the wall the minus signs cancel out and we’re just left with 2 * m * VX divided by the quantity 2L over VX and simplifying this we find that the force experienced by the wall due to this one Collision from this one gas molecule is mvx 2 divid L that is the mass of the gas molecule times its speed along the X Direction squared divided by the length of the wall along the X AIS but that’s just one gas molecule pressure is the sum of all such forces added up across all constituents in the ideal gas and then dividing by the area of the wall in question so what we really want is the total force exerted by all collisions by gas molecules on the wall in a given time and we want to divide that by the area of the wall which is just l^2 well the total force will be given by adding up the forces exerted by individual gas molecules with their individual velocities the component along the x-axis so for instance there’s there may be avagadro’s number worth of ideal gas constituents and so we have to look at each one in a Time window delta T during which these Collision should be considered that time window is given by 2L over V and we find that all we have to do is sum up M V1 x^2 over L Cub plus mv2 x^2 over L Cub plus mv3 x^2 over L Cub all the way up to the total number of molecules that make up this gas notice that every term in this sum has a common multiplicative factor of M the mass of the constituent divided by L cubed effectively the volume of the container so we can pull that out in front of the sum and then we just have to sum over this velocity squared of all of the gas molecules along the x-axis well the gas molecules are colliding with each other we looked at this in a simulation so they don’t all have the exact same horizontal speed at any given time but they do collide with each other and they do on average have the same speed over some unit of time so what we can do is we can approximate this sum by saying that we’re going to consider the fact that all of the gas molecules have on average the same horizontal component of velocity and that sum will just then be given by the total number of molecules capital M times the vx^ S average the average of the square of the X component of their velocities that number is one thing for all of the gas molecules even even if each of them has a slightly different horizontal component of speed because they’ve been colliding with each other and with the walls simplifying this one step further we can replace Big N the total number of gas molecules by avagadro’s number na times the number of moles of the gas little n that appeared in the ideal gas equation and that’s why we’re putting it in here so the final equation we get is that the pressure exerted by the gas on the wall is just on average given by the mass of each molecule or atom divided by the volume of this cubic container times the number of moles of the gas times avagadro’s number which tells you the number of things per mole times the average of the square of the X component of the Velocity well let’s see if we can relate that X component to the total speed of each gas molecule on average on average the X component of a constituent’s squared speed will simply be 1/3 of its total squared speed v^ s the speed of a single molecule will just be given using a variation of the Pythagorean theorem as the sum of the squares of the components vx^ 2 plus V y^2 plus vz^ 2 so on average we would expect after any number number of collisions that each of those components will be 1/3 of v^2 so if we plug that in we take our pressure equation which is just Rewritten here and we plug in the fact that vx2 average is really just 1/3 of the average of its total speed squared we finally arrive at a situation where we can begin to relate microscopic properties like the average speed squared of molecules and their masses to the large scale properties of the whole gas for instance multiplying this equation by the volume cancels out the v in the denominator of the microscopic equation we wind up with P * V is equal to the mass of each constituent times the number of moles of the constituents times avagadro’s number times the average speed squared divided 3 well we can simplify this further by remembering that we defined molar mass the mass per mole of the ideal gas and that’s just given by the mass of each constituent time avagadro’s number so that replaces M and na in the equation and we wind up with the molar mass times the number of moles times the average of the speed squared of a molecule divided by 3 well by the ideal gas law PV which is equal to this thing is also equal to nrt notice that the number of moles of gas appear on the right and left of this equation and cancel out and we can actually finally solve for the average speed of a single molecule in an ideal gas by rearranging this equation to isolate V average and when we do that we find out that this microscopic property of an individual gas molecule its average speed is given by a combination of the macroscopic properties of the gas the square root of three times the gas constant which is just a number times the temperature of the gas divided by the molar mass of that gas the microscopic has been connected to the macroscopic we see here that classical physics can give you some insights into how the individual constituents of a material have relationships with the macroscopic properties of that material that are easier to measure on the human scale we can take one final step and instead of looking at just the speed or average speed of an individual gas molecule we can consider the average kinetic energy of any single constituent of the gas system well that’s just going to be equal to 1/2 times the mass of a constituent times its average speed squared that’s the definition of the kinetic energy of a typical molecule in the gas now from the ideal gas relationship between average speed temperature molar mass and the gas constant we learn the following that the average kinetic energy of a single constituent in the gas which is given by 1 12 m v average squared can be instead related to the macroscopic properties of the gas2 * m time the quantity 3rt / the molar mass now this can be further simplified by replacing the molar mass instead with the mass per constituent times avagadro’s number which is also just a constant and we notice that the individual constituent masses vanish from this equation and we are left with the following that the average kinetic energy of a constituent of an ideal gas is given simply by a number three halves times another number the gas constant divided by avagadro’s number times a single variable the temperature of that gas now it turns out that R the gas constant divided by avagadro’s number is actually related to another fundamental constant of nature which is known as Bolton’s constant it’s written as a lowercase k with a subscript B and so in the end we find out that the average kinetic energy of a single constituent of a gas regard less of the masses of the constituents of that gas is simply given by three halves time the boltmon constant times the temperature of that gas this is a remarkable observation a fantastic relationship that something so tiny as the kinetic energy of a typical thing inside of a vast number of gas molecules is related to this singular macroscopic property temperature that we can can control easily in the macroscopic realm now boltzman’s constant is given here as 1381 * 10 -23 JW per Kelvin it’s a very tiny number which makes sense because the average kinetic energy of a constituent of a large number of gas molecules ought to be a very tiny number even for a standard temperature at room temperature for instance now when we measure the temperature of an ideal gas what this tells us is that we are actually measuring probing in a very direct way the average kinetic energy of its individual constituents and this tells us what heat energy is heat energy is determined by this thought process to be related to the average kinetic energy of constituents of a material body that is to say as one adds heat energy to a system this raises the average kinetic energy of the constituents adding heat Q raises T temperature and this proportionally results in an increase in the average kinetic energy where is the Heat Going the heat is going into the kinetic energy of the individual gas molecules if you want to remove heat from a system all you have to do is find a way to reduce the average kinetic energy of the constituent of that system this also allows us to finally understand that a system with no kinetic energy that is constituents that are holding perfectly still experiencing no collisions with the walls of their container or with each other because there’s no motion at all that is identified as being the lowest temperature that you can ever have zero average kinetic energy for your constituents is 0 Kelvin we finally have a physical understanding at the most basic microscopic levels of a large system as to what it means to achieve zero temperature zero temperature a state of zero heat energy is also a state of zero average kinetic energy for the constituents of that system so this raises an interesting question then how do you transfer heat energy either to or from a system well there are many ways to do this and I’m going to focus on three quite broad established mechanisms for transferring heat energy from a system because ultimately I only really want to focus on one of them so let’s consider cooling heating will just be the reverse of any of the things that I say here let’s begin with the mechanism of conduction conduction is when you place a second system perhaps at a lower temperature if we wish to cool the first system in physical contact with the first system think of two cubes of metal at different temperatures we want to cool one of those blocks of metal so we take another system that’s even cooler and we press them together so that their two faces of the material are physically touching each other at that interface at that cont space between the two materials collisions are going to begin occurring between the atoms or molecules of one system and the atoms or molecules of the other system this creates a an arena in which collisions occur transferring kinetic energy from one system on average to the other what you’ll find is that higher kinetic energy constituents are going to typically lose some kinetic energy to the slower moving constituents at the interface of the other system of course at the interface of the other system those constituents will then start having more collisions with the things inside the system and that’s how heat energy is transferred by conduction through a system it’s all collisions this decreases the temperature of the hot system and increases the temperature of the cold system until such time as the temperatures of the two systems reach a new equilibrium position T1 equal T2 this will will occur typically when the temperature of the hotter system is lowered down and the temperature of the cooler system is raised up and you finally reach a point where they both have the same temperature and they stop transferring heat energy they on average have the same kinetic energy for all their constituents no more transfer can occur then there is convection in convection you pass a fluid like a gas or a liquid across or around another system so if we want to take a system and cool it we might blow air over it or push water across it in some kind of current collisions at the boundary of your system between the constituents of your system and the constituents of the fluid will transfer kinetic energy on average to the fluid the fluid if it’s cooler will have um lower kinetic energy constituents and collisions will tend to favor increasing the kinetic energy of the cool systems uh constituents and this ultimately cools your target system system one by lowering the average kinetic energy of that system and finally there’s radiation radiation is a process by which constituents lose energy by giving it up in the form of radiation of light for instance you might be familiar with the fact that you can stretch your handout several centimet inches maybe even up to a few feet away from a a h hot cooking pan on the stove and even though you are not making physical contact with that and even though the air is very still in the room around you you feel something being transferred to your hand you would say that you can feel from a distance that the pan is hot well that’s because it’s radiating typically at the infrared and that infrared radiation which you can’t see with your eye but which you can feel with your skin uh will be absorbed by your skin radiation requires no physical contact between a system and the environment in fact if you took all the air out of the room and stuck your handout in that environment you would nonetheless feel heat being transferred to your hand by radiation electromagnetic radiation requires no medium to travel and so even evacuating the room of air will still lead to a cooling of the pan in this case by the radiation of infrared light now radiation has the effect of carrying kinetic energy away from a system and giving it to the environment large around it even without physical contact radiation is what I’m going to focus on for the rest of this lecture it’s an interesting phenomenon because it is an interface between mechanics and electromagnetism and you can already begin to see that since we got ourselves into trouble thinking about motion and the laws of electromagnetism and the laws of mechanics that a place like this heat energy and radiation is another similar interface of classical mechanical view of the universe with the electromagnetic laws of nature where inconsistencies may arise if you overly trust the mechanical laws of nature there’s a mathematical relationship that has been determined by experiment in the late 1800s and early 1900s between the energy that is emitted or absorbed by a heated material body and the temperature of that body this was determined empirically by ysep Stefan to be the following that the power radiated or absorbed by a body that that is to say the change in heat energy per unit change in time is given by the product of four numbers Sigma which is a constant of nature known as the Stefan boltzman constant whose value is 5.67 * 10-8 watts per meter squ per Kelvin to the 4th it’s not a bad number to remember because it’s got 5 6 7 8 in it I find that handy for remembering this number in a pinch now the stuff on boltzman constant is multiplied by another number which is this curly lowercase Greek Epsilon Epsilon is the emissivity of the surface of a body and it ranges between zero no emission and one perfect emission you can see that a body with zero emissivity will emit no power in the form of radiation because the right side of this equation will always be zero on the other hand a body with perfect emission will maximally emit radiation given by the product of the other numbers the Stefan boltzman constant the surface area a of the body and the temperature of the body raised to the fourth power note that all material bodies above 0 Kelvin radiate energy in the form of a electromagnetic radiation you and I sitting here right now at 98° F which is the typical human body temperature are radiating light away from our bodies we just can’t see it and we can play around and figure out what wavelength it is as an exercise in class A Perfect emitter with emissivity of one is also known as a black body it’s a very special kind of object it is a system that absorbs all incident radiation and it can subsequently re-emit its own radiation with perfect emissivity black bodies are a special laboratory for testing the interface of the laws of mechanics the movement of the constituents and the laws of motion that describe the allowed states of motion of that material at its smallest level and electromagnetic radiation the emission of light now before I show you an example of how classical physics when applied to the question of radiation got it wrong I want to Define for you a very useful concept and that is the power emitted per unit wavelength in a radiation situation this is known as the spectral Radiance now in a situation where an amount of energy say Delta Q is radiated Away by a body in some some period of time delta T it is actually fairly typical to ask the following question to really drill down into a question about the amount of energy within a certain range of wavelengths or frequencies of the emitted radiation in other words if I consider a range of the radiation with a minimum wavelength Lambda and a maximum wavelength that’s just a little bit higher than that Lambda plus Delta Lambda where Delta Lambda could be a very tiny amount how much energy per unit time is radiated by wavelengths in that range and asking this question is answered by a special kind of function known as the spectral Radiance now it’s often denoted by various letters I’m going to use the capital letter b and I’m going to make it a function of Lambda the wavelength explicitly to emphasize the fact that it is answering a question per unit wavelength this is the energy radiated per unit time per unit wavelength I could have also alternatively written B in terms of the frequency F because frequency and wavelength are related to the speed of light for electromagnetic radiation but I’m going to use b as a function of Lambda if you want to know the power radiated around a specific wavelength then you need to pick a small range around that wavelength and compute the product so for instance you might choose a specific Lambda and then because this is defined over a small range of Lambda to Lambda plus Delta Lambda you need to multiply the spectral Radiance which is a function of Lambda times the window around which you are trying to compute the amount of power radiated Delta Lambda and that will return the power emitted around that wavelength now that would be a sort of discret way of thinking about it if you have a well- defined continuous function a function of Lambda that varies continuously as Lambda representing this spectral Radiance B then you can just integrate you can use integral calculus in a Range to get the answer you desire so for example if I want to know how much power is emitted between two wavelengths Lambda 1 and Lambda 2 I can simply take the product of B and D Lambda and integrate that product from Lambda 1 to Lambda 2 and if B is a well-defined function I can do the integral it may not be pretty but I can get a function that answers the question and gives me the power radiated in that range of wavelengths now with that introduction in mind let’s take a look at a classical physics attempt to predict the amount of energy emitted per unit time about a given wavelength Lambda this was worked out in the early 1900s and answers the question how much power power is emitted in say uh the ultraviolet range around 240 nanm and some window around 240 nanometers how much power is emitted in the range of red light around say 740 nanometers in some window around that answering that question in little steps through the electromagnetic spectrum will give you a a picture of how power is distributed as a function of wavelength in the emitted r radiation now the classical version of this is known as the genes law and it’s from 1905 and so again you have to start from the spectral Radiance function the power per unit wavelength that is this quantity here in the Ry genes law 8 Pi * a the surface area of the object time C the speed of light times the boltson constant times the temperature of the object divided by Lambda to 4th and if you check the units of that particular fraction you’ll see that it is jewles per second per meter so per unit wavelength if you then want to know in a small window around the target wavelength Lambda how much power is emitted you need to multiply that by the size of the window and that will then answer the question about how much power is emitted around that wavelength in a window about the wavelength Lambda so for example this tells us that for say a spherical body that’s heated to uh a certain temperature T and that body has a certain surface area a the shorter the wavelength of the radiation you consider being emitted from the body the more and more power is radiated around that wavelength if true this would be a catastrophic feature of nature so for example consider a small sphere of metal or something like that you you make it out of a very good material and it’s got a surface area of just 1 M squared and it’s got an emissivity of one if you heat that to 6,000 Kelvin and just for reference a very modest small propane torch can easily heat something to 3,000 Kelvin you would emit about 10 to the 16 Watts that is Jews per second alone in dangerous ultraviolet radiation for instance with a wavelength of 250 NM that is easily lethal to a living organism to give you a point of reference you can buy easily on Amazon or at other online vendors a sanitizing wand A sanitizing wand emits 4 watts of radiation power in the form of ultraviolet specifically ultraviolet C which has a wavelength which kills bacteria now if it can kill bacteria it can do significant damage to other kinds of living cells including the cells of the human body body you should never expose your body to UVC if you can avoid it because it causes damage to DNA and this can lead to the formation of cancers 10 the 16th watts of UVC would be extremely dangerous if not lethal and all from a small heated sphere at 6,000 Kelvin well that seems ludicrous and it is ludicrous if you actually go and measure the amount of power emitted at a given wavelength it doesn’t shoot off to Infinity as Lambda goes to zero this is just not what is observed in reality and yet it is a byproduct of thinking of classical physics the marriage of Newton’s mechanics with electromagnetism let me show you a graph I don’t want you to worry too much about what the axes mean I’m going to describe them in an oversimplified manner the vertical axis tells you how much energy is emitted per unit time per unit area and per unit solid angle so at some chunk of uh angle space for a given frequency of radiation you’re considering so the frequencies are on the horizontal axis high frequency corresponds with short wavelength ultraviolet radiation would have a shorter wavelength x-rays would have a very short wavelength and so forth on the other hand long wavelengths are down here at low frequencies so infrared and red they tend to have very small frequencies and correspondingly very large wavelengths the blue curve which not only comports with reality but was predicted in a mathematical exercise by a physicist named Max plunk that one is what nature should look like and in fact is what nature does look like if you heat a black body to 5800 Kelvin and look at the so-called spectrum of emitted power for a given frequency the blue curve is what nature looks like this yellow dotted curve is the prediction of the Ry jees law and comes nowhere near reality it arguably maybe does an okay job for the very lowest frequencies the very longest wavelengths of radiation from a body maybe a human body would be accurately described by the Ry genes law but the sun on the other hand which has a temperature of about 5800 kin also behaves like a black body and is nowhere near described correctly by the Ry genes law now another physicist named Vil heline figured out in 1896 his own version of this prediction and that’s the pink curve and you’ll notice that ven’s law as it’s known does a pretty good job of describing the radiation at the highest frequencies but does an abysmal job of describing radiation at low frequencies Plank’s law however Nails it Max Plank’s law as he derived it in the early 1900s was the Cornerstone of the correct description of the radiation from heated matter so you can see here again a place where there’s a breakdown between classical thinking motivated by the things that we learn in introductory physics the things that are from The Familiar macroscopic World applied to the world of the very small in this case the individual constituents of a heated body of matter there’s a breakdown here and a breakdown is an opportunity to make sense of the correct laws of nature Max plunk figured it out even where V and Ry genes could not so to review in this lecture we have learned the following things we’ve learned about the connection between temperature and the constituents of a material body we’ve explored the precise nature and cause of heat energy the fact that heat energy is related to the average kinetic energy of the constituents of material like an ideal gas and that that is directly related to the temperature of the macroscopic body of that gas we’ve considered ways of transferring energy to and from objects and we’ve looked specifically at the emanation of electromagnetic energy in the form of light from a heated body we’ve looked at some of the laws that were either derived or determined to govern that kind of radiation of energy and we’ve seen that in places where classical physics mechanics Newton’s laws were combined with electromagnetism to predict the radiation from a

heated body a special kind of body a black body is a total breakdown compared to reality in the next phase of the course we’re going to take this breakdown as a la launching point for a deeper understanding of nature we’re going to transition from the very fast to the very small and begin to explore the origins of quantum [Music] physics in this lecture we will learn the following things we will learn how the black body radiation Spectrum was finally understood we’ll learn about the possibility that’s implied by that solution that energy may come in discrete units we’ll learn about a phenomenon known as the photoelectric effect and we’ll learn how Albert Einstein resolved the puzzle of of the photoelectric effect in the last lecture we saw how the ra Gene power Spectrum prediction utterly failed to model nature correctly given a black body heated to a certain temperature T the rayy genes model predicted that more and more energy should be emitted in shorter and shorter wave lengths leading to some kind of natural catastrophe merely heating up a body to a few thousand de Kevin however matter heated to a temperature T simply does not radiate according to that prediction because if it did the effects would be catastrophic the shorter the wavelength and the more damaging the electromagnetic radiation the more of it would have been emitted from such a body as predicted by the Ry jees model it simply did not comport with reality this mismatch between reality and the prediction of classical physics has been called the ultraviolet catastrophe correspondingly the mismatch between reality and the prediction of ven’s model is known as the infrared catastrophy now historically this problem was not considered threatening or really so important that anyone truly panicked although at least one individual T did take this problem extremely seriously and that’s Max plunk you got to admit though ultraviolet catastrophe is a lovely and exciting name um it I should note that the term ultraviolet catastrophe actually doesn’t date to the exact period when the Ry jeans prediction or plun uh and his work were established but it actually appears to date to much later about 1911 and seems to have been coined by the physicist Paul Aron Fest now in the last lecture we saw that there was a model by a man named plunk Max plunk that did seem to have gotten the right answer so what was it that Max plunk did well he started from a mathematical model of a perfect black body a simple Model A a cavity fully enclosed on all sides except for a tiny hole in the cavity an ideal black body to remind you is one that absorbs all incident radiation on it and then it re-emits its own radiation with some Spectrum it’s got perfect emissivity so it maximally radiates given its other physical properties now once one hypothesizes that such a system exists one then has to apply the laws of physics to predict or describe that emitted radiation Spectrum the amount of energy emitted for instance per unit solid angle per unit time per unit wavelength and per unit area uh per unit many things but your bottom line is you’re attempting to predict how much of each wavelength interval of radiation is present in the emitted bulk of radiation now a cavity with a single small hole in it is actually a really good model for a perfect black body if you shoot radiation at the hole 100% of it incident on the hole will enter the cavity and be lost to the outside world that radiation is absorbed by the cavity now it then enters the cavity and it begins bouncing around inside the cavity striking the walls and therefore hitting the bits of matter that make up the walls of the cavity and fundamentally as we’ve seen in physics 2 matter is made from electric charges now as we also know as these electric charges get struck by radiation they’re going to begin to gain kinetic energy which will cause them to heat up the material surface of the cavity inside the cavity a hotter object emits radiation in a different way than a cooler object so again the question we want to boil this down to is what will that spectrum of emitted radiation due to the heating of the walls of the cavity from the incident radiation actually look like now we can boil the black body problem down to just a very simple collection of phenomena that we can conceptualize of using information from physics 2 uh this is a very simple model of an electromagnetic wave which would be what the radiation impinging on the surface of the cavity walls would look like it’s got an oscillating electric field and perpendicular to it it’s got an oscillating magnetic field and it’s traveling perpendicular to both of those fields this wave then strikes a charge in the wall of the cavity so for instance an electron the electron feels the electric and magnetic fields of the wave and it will respond to those by accelerating this is what we learned in physics 2 The Wave with its increasing and then decreasing electric field strength for example will cause an electron to accelerate more than less it will oscillate it will wave like a bit of matter in a rope that’s wiggled or in a chain That Shook or in a string that’s plucked the electron will oscillate so radiation enters the cavity with any number of possible frequencies or wavelengths that can compose that incident radiation and all of it is taken in by the cavity through the hole the electric charges that make up the matter in the walls of the cavity will either scatter they’ll be knocked off of their parent atoms for instance or maybe they’ll wiggle in response to the electromagnetic wave that strikes them and thus absorb some of the electromagnetic radiation as motion now absorbing an electromagnetic wave causes the charges to oscillate and an oscillating electric charge is a source of an electromagnetic wave so these newly oscillating electric charges can emit their own electromagnetic radiation this is the source of the emission spectrum from the black body so what will that reradiated energy look like when it escapes the cavity that radiation too will bounce around inside the cavity but some of it will make it out of the hole what will it look like and how much of each frequency is found in its power Spectrum well recall that the rayy genes model using a purely classical model of all of this system mechanics and electromagnetism via Maxwell’s equations deter that the Spectrum should look something like this that the energy emitted per unit time um taking into account the surface area and the whole uh viewing solid angle of the of the black body will basically go as the temperature of that body over the wavelength to the fourth of a particular wavelength of light that we’re considering as part of the outgoing Spectrum but as we can see as you decrease the wavelength that is increase the frequency of the radiation more and more and more power is emitted by the black body now a key assumption that lay underneath the building of the ra genes model was that all frequencies are possible for oscillating charges a charge stuck in an atom in the wall of this cavity model can oscillate at any frequency it likes all frequencies are possible and that led to the ra gen model let’s make a very simple model of a system where we can cause oscillations to occur in electric charges and then those oscillating electric charges in turn emit electromagnetic radiation radio waves or light that light then travels across a gap striking another electric charge and setting it into oscillatory motion to illustrate what I mean by this imagine we have the ability to wiggle an electric charge over here at a transmitter site and watch a sympathetic wiggle over here at a receiver site when an electromagnetic wave from the transmitter reaches the receiver to illustrate this let me start oscillating the electric charge on the left what you’re seeing here is the full electric field around that charge as it changes in time as the charge Moves In Space the changing electric field propagates out at the speed of light and causes an oscillatory pattern in space some places have strong electric Fields pointing in One Direction some places have weak electric Fields some places have electric fields that point in the opposite direction we can better see this by looking at the amplitude of the electromagnetic wave as a function of position away from the o oscillator and we see the rising and falling in time of the wave as it travels to where the receiver is the oscillator in this model is a charge that has been set in motion by radiation that was absorbed by the cavity walls the absorption of the radiation causes the charge to oscillate and the oscillating charge in turn emits its own electromagnetic radiation so we’re watching a charge that’s been set into oscillatory motion by external radiation emitting its own radiation here on the left and then causing another charge to oscillate over on the right that would in turn of course cause that secondary oscillation to generate its own radiation and you can see how the black body problem is a very complex interplay not only of mechanics but electromagnetism and getting the details of this right are essential to correctly predicting the radiation from a black body now the Fatal flaw that people like ra and genes made when constructing their prediction for the energy emitted per unit solid angle per unit time and per unit wavelength from say a black body was that they assumed that any oscillatory frequency was possible for the charges it seems a natural assumption electromagnetic waves originate on oscillating electric charges if I change the frequency and I can change it to anything I like in classical physics I expect a different kind of electromagnetic wave with its own frequency to be emitted and In classical physics I can pick any frequency I want anyone at all because in classical physics they’re all possible they’re all allowed and this was the Fatal flaw it turns out in the the Ry genes calculation of the black body spectrum they assumed that those oscillating charges in the walls of the cavity could emit any frequency of radiation they wanted as they sympathetically begin to oscillate having been struck by external radiation it turns out that this leads to the Ry Gene’s prediction of the power Spectrum which is utterly wrong the plun model on the other hand which arrived at the correct answer results in a power spectrum that looks like this it goes as 1 over Lambda to a power in this case Lambda to the 5th but there’s an overall multiplicative factor and that’s where the temperature dependence shows up it’s also where the where a wavelength dependence shows up as well and this extra piece has the effect of cutting off the power Spectrum at high frequencies in other words as you go to higher frequency you actually see there’s a turnover in the prediction of the model and it drops off to zero as you go to shorter and shorter wavelengths higher and higher frequencies you don’t emit more energy you wind up emitting less now what was the difference between the rayy Jee model and Plank’s effort to model the black body Spectrum well one key assumption was that plunk did not allow all frequencies to be possible for us oscillating charges and I’ll return to that assumption in a bit looking at some of the historical context of Plank’s own work to give you a better sense of what atoms and molecules actually do when they are struck by electromagnetic radiation let’s look at this simulation incorporating the modern understanding of the interaction of radiation and Mattern we have here a a water molecule two hydrogen’s bonded to one oxygen and we can shoot radiation at it let’s begin by shooting microwaves long wavelength electromagnetic waves somewhere between visible light and radio if we start shooting microwaves at the water molecule we see that many of the microwaves will pass through the water molecule but some of them will be absorbed and cause rotational motion of the molecule which then scatters the microwave this is in fact how a microwave oven Works microwaves at the right frequency will cause water molecules to rotate and collide with each other and kinetic energy is added to the system and as we know kinetic energy is related to the temperature of material if you add kinetic energy to the water molecules in a system you will heat it up let’s change the wavelength of the radiation to infrared we are now shooting much shorter wavelength light at the water molecule no longer are we able to make it rotate rather we are able to make it oscillate the hydrogen atoms that are bonded to the oxygen will occasionally be struck by an infrared Photon that then causes them to jiggle around a little bit before scattering off the photon if we shoot visible radiation which is even shorter wavelengths at our water molecule we see that it is effectively transparent to the visible light all the visible light all the visible light radiation is passing through the water molecule as if it’s not even there and that shouldn’t come as a surprise to us water is transparent to light so it makes sense that visible light should be able to make it through a body of water and we see that modeled here if we shorten the wavelength of the radiation even more to alter ultraviolet we see that this also tells us something about water that water doesn’t respond to this wavelength of radiation ultraviolet radiation passes through the water molecule essentially unscathed this kind of little simulation incorporates our modern understanding of electric charge chemicals bonding and the ways that energy can and cannot be absorbed and reiated by atoms and molecules we see that not all radiation causes a water molecule in this case to do anything only certain light frequencies or wavelengths have an effect on the charges of the water molecule and thus can cause them to vibrate oscillate or rotate in such a way that that might result in subsequent later reradiation of energy now as I showed you in the last lecture video this model accurately describes the shape of a black body Spectrum but it comes at one small cost Plank’s effort resulted in the need for a new physical constant which he labeled H and eventually came to be known as plunk constant it is related to the degree of the discretization of the oscillation of the charges in the cavity in other words not all frequencies of oscillation are allowed and H tells us something about the gap between allowed frequencies things in between in the gaps are not allowed this is known as the quantization of the oscillatory motion of charges in the cavity walls quantization coming from Quantum a Latin word for how much implying not an unlimited set of values that are possible for a system but rather a discreet well-defined and finite set of values that are are allowed for a system with no values in between the allowed ones now the reason that the Spectrum winds up cutting off at Short wavelengths or high frequency is that electromagnetic radiation as a consequence of Plank’s model requires a specific amount of energy to make a specific wavelength in other words if you want to make ultraviolet light you’ve got to put in a minimum amount of energy to do that if you want to make something with a shorter wavelength than ultraviolet light like xray Rays or gamma rays you have to put in even more energy and not all of those energies are possible inside the oscillating charges of the cavity walls so if you don’t have that energy you can’t make that wavelength and the Spectrum naturally Cuts itself off this implied also that the energy of the radiation is quantized and itself can come in units or packets now now this new constant Plank’s constant H ultimately had to be determined from experiment it wasn’t predicted by Plank’s model it was a parameter in the model that had to be determined and it has units of Jews time seconds which if you flip back to physics 1 and play around with those units a little bit you’ll realize that they correspond to units of angular momentum this actually has deep implications for the universe but we’re not going to get to them right now now its value was originally determined by Max plun by simply changing the value of H around in his calculations until at a specific temperature for a black body he had a value that yielded a shape for the black body spectrum that best described that particular heated black body now that’s how he did it and in fact by doing this by fitting the parameter to the data and determining the value of the parameter itself he came to within a few percent of the currently accepted value of plunk constant which is already a remarkable achievement but in science if you build a model by tuning it to existing experiment the true test of a model is where whether or not it correctly predicts new phenomena that have not yet been either explained or observed so plunk constant by itself being determined from the black body may just be tuned a mathematical model to the data to get the answer you wanted in the first place that’s the first step in describing nature but if you want to see whether or not you’ve learned something deep about nature you need to find the next thing that you can test by applying the same idea with the same constant and see if you get answers that are consistent with nature now the currently accepted value of Plank’s constant is 6. 626nightmarket time 10 -34 Jew seconds that is a number worth memorizing on par with the speed of light 2.98 * 10 8 m/s were the mass of the electron 9.11 * 10us 31 kg Plank’s constant is one of those fundamental numbers that when committed to memory can be busted out when you need it to do a quick calculation and can be very handy when doing things like engineering new systems like in electronics for instance now this constant is crucially important in the modern world I I can’t understate its value any more than I can understate the value of the speed of light its value is now the basis of the system international definition of the kilogram the definition of the kilogram used to be based on the size and mass of a platinum aridium bar that was kept under glass in France there are many flaws with that for instance if atoms of that bar flake off over time and you don’t notice it then over time your definition of the kilogram using that as a reference changes weights and measures are crucial to things like economies and standards and so forth and so you don’t want your definition of the kilogram drifting over time now so far as we know Plank’s constant is stable over vast periods of time certainly over many billions of years and so it was wise to redefine the kilogram using something that itself can be determined independently and as stable and it turned out that a particular way of measuring plunk constant lends itself to defining the kilogram and that change went into effect only in 2018 plunks constant also plays a fundamental in key role in all electronic devices certainly all modern micro Electronics those devices rely on the exact properties of semiconducting materials and semiconductors can be precisely engineered thanks to the quantization the discretization of radiation and matter and ultimately all of this stems in its scale size and control from the value of Plank’s constant now as I’ve hinted before Plank’s work had a consequence built into it that if true would radically change our view of radiation electromagnetic waves he realized in his paper on the subject that as part of the only way he could find to describe the black body Spectrum he was forced to assume that radiation had to come in quantized units whose sizes were controlled by the constant H and proportional by that constant to the frequency of the electromagnetic waves this equation relates the energy and the frequency of electromagnetic radiation E equals H Plank’s constant time F the frequency of the radiation and since fre fre quency and wavelength are related by the speed of light this also implies a relationship to the wavelength of that light let me give you some of the context of Max plunk and his work he concluded this effort in 1900 after many desperate years of working on the problem but he himself did not fully accept the implication of what his newly developed constant H implied and the consequences of his solution to the black body Spectrum problem basically his solution implied if correct that matter and energy can be quantized into discrete units and that units in between those are simply not realized in nature they’re forbidden by the system somehow by the parameters of the system now he assumed that this was all some kind of convenience math trick that he had played that it wasn’t really describing nature at a fundamental level and that someone else would come along really solve this problem using the correct description of Nature and one day explain why the trick worked if you look at some things that plank himself has said over the history of his own life from the year in which he published his black body Spectrum paper to decades later as he reflected on that period of his life you can gain some insights into his psychology as a scientist at the time and in the paper that he published in 1900 he States moreover it is necessary to interpret the total energy of a black body radiator not as a continuous infinitely divisible quantity but is a discrete quantity composed of an integral number of finite equal parts you can see here in sort of the tone and writing of his sentence that he finds something necessary to do but he doesn’t necessarily take away from that that it implies reality follows from this assumption the assumption that the total energy of a black body radiator is discretized and not continuous May merely be a mathematical assumption but nonetheless he found it necessary to make this assumption in order to interpret the data now many decades later in a letter that he wrote to RW wood he reflected back on this period and one famous quote from this letter is often repeated wherein he said the whole procedure was an act of Despair because a theoretical interpretation had to be found at any price no matter how high that may be and you get a real taste of his professional desperation where others had failed to describe the black body Spectrum plank was desperate to figure out what Avenue would lead to to the correct description he didn’t necessarily accept that the mathematical steps required to follow that Avenue implied anything about nature but it worked and he published it even if he didn’t fully embrace the implications of his own work now another famous quotation from Max plun whose Source I simply couldn’t track down but it is attributed to him by many other sources was that he was ready to sacrifice any of his previous convictions about physics in order to solve this problem now this last quote especially was motivated by another thing that plon had to do to solve this problem and that was to employ a statistical description of matter and radiation um many physicists found statistics distasteful because under the hood statistics tells you that you can’t know for sure the outcome of a particular system but you can know the probabilities of all possible outcomes even if you don’t know which one will be realized in the next experiment many physicists who believed that the Universe was deterministic that is that if you know exactly the initial conditions you can find the exact outcome of the system every time found the use of Statistics to describe nature distasteful distasteful doesn’t necessarily mean wrong and that’s why the hard work of the scientist is to use obser ations of nature to assess the assumptions that we have made in trying to describe and predict nature now as I said before the burden in science of a new idea Falls not on your ability to describe the things that came before but to explain the things that come after without changing any of the assumptions of the idea a truly successful Theory a theory that is not not only built on facts but predicts the existence of new ones is ultimately forged in the fire of experimental science married with mathematical effort this lands us on the subject of the photo electric effect now the photoelectric effect was known in the late 1800s but could not be described using what was known in the late 1800s it was observed by physicist Heinrich Herz now he was the first person to definitively demonstrate the existence of electromagnetic waves these had been a phenomenon predicted by Maxwell’s equations and in that same prediction captured the essence of light that light itself is an electromagnetic wave Herz realized that if you were going to test the prediction that electromagnetic waves are real independent of light you would need to demonstrate their existence by transmitting them from one place in a laboratory receiving them at another and showing that the wave induces an oscillating electric charge at the Target location so what he ultimately showed was that an oscillating charge at one place in a room a laboratory could induce an oscillating charge elsewhere in the room with no physical contact and this established the reality of electromagnetic waves Beyond light in fact you could think of this as the first radio transmission now he was also the first person to demonstrate an intrigu physical phenomenon the photoelectric effect light which is an electromagnetic wave at heart at least in the Maxwell view of nature um shown on a metal can liberate electrons from the metal so take a beam of light shine it on the surface of a metal look for an electric current and under the right conditions you will see an electric current develop in the metal now Maxwell’s equations predict that the intensity of a light beam an electromagnetic wave is proportional to the squared strength of its electric field that is e squared if e not is the base maximal electric electric field value of a particular wave now because of that prediction uh attempts were made to describe and predict and explain the observed features of the photoelectric effect so let me use an analogy combining mechanics and the laws of electrom magnetism Maxwell’s equations to attempt to predict the set of phenomena that you would expect to arise in the photoelectric effect think of the charges in a metal as a ball that’s stuck in a pond in a patch of lily pads or weeds what you want to do is Liberate the ball you would like to knock the ball out of the lily pads free it so that it floats over to the shore and you can get it cuz you don’t want to step in all of these weeds who knows what’s swimming around in this thing fine so you and your friends devise a sort of classical photo El electric effect experiment you get a bunch of empty buckets that you might have around to keep ice you know keep your beverages cool while you’re playing that day you empty out the buckets and you you carry them over to the shore of the the pond and uh one of you kneels down at the edge of the pond and starts using the bucket to push on the surface of the pond well this generates water waves so you’re pushing on the surface of the pond and the water waves are making the ball and the lily pads wiggle up and down but it’s not knocking the ball loose no no problem you’re at the limit of your strength but you’ve got lots of friends so your friends all also kneel down at the edge of the pond near you and they start pushing on the surface of the pond and you’re not very coordinated so these waves have different amplitudes at different times but eventually if you’re patient enough some waves will pass through the ball they’ll add up an amplitude constructively interfering and they’ll deliver enough energy to the ball to knock it out of the lily pads so the photoelectric effect in analogy to this ball stuck on a pond in a bunch of lily pads uh should be behaving as follows if you send in light waves even feeble light waves that don’t themselves have enough energy to liberate a charge from a metal if you send in enough of those light waves at the metal you will begin liberating charges the light wave amplitude should add up they go as the electric field squared of each wave and if you wait long enough you’ll start knocking electrons out of the metal that’s what people expected from the classical theory of mechanics and electromagnetism but what was actually observed in the close study of the photoelectric effect well what was observed was that the intensity of the light you shine on the metal has no effect on initiating the effect itself the photoelectric effect can’t be induced by simply cranking up and up and up and up the intensity of light if that light doesn’t already seem to have the ability to make a current flow in the metal we can simulate The observed photoelectric effect using this fet simulation that’s available on the web for example I can start by trying to shine long wavelength light onto a metal I’ve selected a copper plate which is located on the left side of the apparatus I have a representative light source at the top of the apparatus and as you can see I can control the intensity of the electromagnetic radiation or light that I can Shine On The Copper I’m going to go ahead and crank this red light source up to 100% of its intensity and as you can see there is no observed current in the graph on the right the graph shows on the y- AIS the electric current that’s observed in the system and on the horizontal axis the intensity of the light which is currently pegged at 100% even if you wait 1 minute 10 minutes 100 minutes you think you’re allowing the amplitude to build up and occasionally knock an elron out of the metal but you see nothing now instead if you change the wavelength or frequency of the light you maybe can see what happens to the effect do you make an electric current flow well if you start from a particular wavelength of light that doesn’t cause the photoelectric effect and then you change it gradually to a longer wavelength say start with red light and then change it to microwaves or radio you’ll also notice moving the intensity of the light up and down doesn’t cause the photo El electric effect to start but if you shorten the wavelength from the ineffective wavelength to something shorter higher frequency shorter wavelength at some point you’ll suddenly notice that electrons will will begin to flow through and off the metal you can induce the effect as you shorten the wavelength I’m going to begin to lower the wavelength of the light from Red at about 750 NMM down to Orange down to yellow and we still see that nothing is happening I’ve definitely switched to a shorter wavelength of electromagnetic radiation but we still see no current versus intensity on the graph I’m going to continue to shorten the wavelength of the light now we’re into the green we’re approaching light blue or blue now we’re going to the more richer blues and we’re heading toward Violet now I’m definitely down at the shortest visible wavelength range of light and yet the copper is doing nothing and I’m blasting it with 100% intensity but watch what happens when I push this simulation into into the ultraviolet very short wavelength radiation once I cross below a threshold wavelength or frequency for the radiation suddenly electrons begin to get shot off the copper by the light now over here you’ll notice that the current has gone up a little bit on the vertical axis I’m at 100% intensity and I’ve moved up about one tick mark on this axis now once you’ve set the photoelectric effect in motion you might hypothesize that if you crank down the intensity of the light to some sufficiently low level then the waves won’t be able to add up enough anymore and no more charge will flow even before the intensity gets to zero but what you find is that the electric current that you induce in the metal declines to zero as the intensity goes to zero and the electric curve current only goes to exactly zero when the intensity is also zero that is you switch the light off what I’m going to do now is I’m going to go ahead and lower the intensity of the light now you’ll notice that of course the current is decreasing as the intensity of the light decreases I’m still knocking electrons off but not as many and of course if I bring the intensity all the way down to zero then the photo El electric effect switches off there was no point in the intensity and current plot where the effect suddenly switched off before I got to zero intensity in fact if there’s even a little bit of intensity you’ll notice that electrons start boiling off of the copper not many but they come off some very fast and some very slow at a particular threshold wavelength and frequency the photo El electric effect simp simply begins raising and lowering the intensity of the light seems to have no effect on the maximum kinetic energy of an ejected electron even very weak intense light but with the correct wavelength or frequency will rapidly eject an electron occasionally with a high kinetic energy despite the fact that the intensity scales as the square of the electric field strength and shouldn’t more electric field produce more acceleration that’s what all of that stuff from kul’s law and physics 2 and Max equations says should be happening I can lower the intensity down even more down to just 1% of the source and yet nonetheless electrons will come shooting off of this thing with lots of kinetic energy it’s as if the kinetic energy of the ejected electrons from the copper have nothing to do with the intensity of the light but only to do with the wavelength or frequency of that light now I can bring the radiation up in intensity a little bit so that we can see a few more electrons boiling off the metal and what I’m going to do is I’m going to begin to increase the wavelength of the light just very gradually just a nudge at a time at some point we’re going to cross a threshold where the light simply does not have sufficient wavelength to induce the photoelectric effect and it seems that I’ve gotten to it at about 270 nanm now I can go ahead and crank up the intensity to 100% now that I’ve moved just past the wavelength threshold to induce the photo electric effect and yet again we see that intensity does not suddenly cause the photo El electric effect to switch on now you can see how frustrating this must have been for the physicists of the late 1800s this set of observational facts defied explanation using all the battl tested Notions of classical waves and the laws of electromagnetism does this sound familiar does this sound like the moment that led to special relativity because if it does you’re on the right Trail you’ve found a place where the theory of motion and the theory of electricity and magnetism which were largely developed on macroscopic things then CS a new microscopic phenomenon where it utterly fails to make accurate predictions and that smacks of opportunity so how was the photoelectric effect explained well it was our old friend Albert Einstein who cracked the photoelectric effect in one of his 1905 so-called miracle year Publications this was the year that catapulted him into at least physics academic Fame and allowed him to finally secure a faculty position after years of toil at the patent office in Burn Switzerland now to explain the phenomenon Einstein reached back to Plank’s 1900 paper on the black body Spectrum recall that a consequence of Plank’s solution to the problem desperate though the remedy may have been was that light has an energy that’s given not by the intensity of the electric field of the wave but rather by the frequency or wavelength of the light that is e is equal to H Plank’s constant time F the frequency of the light now since the speed of light is equal to the frequency times the wavelength one can substitute into this to get the corresponding relationship with wavelength shorten the wavelength increase the energy increase the frequency increase the energy those are the relationships between frequency and wavelength and the energy of a light packet a light quantum so Einstein Embraces the implication of Max Plank’s work that radiation can be quantized into discrete units and therefore a single unit of light is hypothesized to carry or cost to produce H * f for light of a certain frequency F so even if one unit of light of a certain frequency strikes an electron and therefore 4 strikes it with a certain amount of energy given by H * F The Liberation of the electron is immediately possible independent of the intensity of the light more light Quant striking more electrons per second means more electric current fewer light quanta striking fewer electrons per second means less current but if you have even one you will liberate a charge and that’s consistent with the observations of the photoelectric effect that once you make it happen it happens happens all the way down to even very low intensity until you switch off the light source so what are the equations describing the photoelectric effect that were worked up by Einstein in 1905 so he reasoned that it takes a certain minimum amount of energy to remove an electron from a metal a metal isn’t just going to give up its electrons without a fight I mean otherwise it would be really easy to uh just reach out and strip electrons off a metal but it takes energy so there’s some minimum amount of energy that’s required to liberate one charge from a metal and this is called the work function and it’s denoted by the lowercase Greek letter fi now if a Quantum of light with a given energy strikes the electron and has energy that exceeds the work function then it’s possible to transfer energy to the electron and remove it from the metal it can scatter the electron or even be fully absorbed by the electron the maximum amount of energy that can be transferred to the electron by such an interaction of matter and light is given by the following equation that the maximum energy that an electron can get when struck by a Quantum of light is given by the energy of the Quantum of light minus the work function it takes some energy to remove the electron if there’s extra energy left over after that it goes into the energy of the electron in motion and finally we arrive from Plank’s hypoth ois about the energy of these light quanta at the equation that the maximum energy you can transfer to an electron removing it from a metal is HF minus 5 now if HF is less than F the electron can’t receive sufficient energy to be removed from the metal HF must equal or exceed the work function in order to liberate an electron from a metal and metals of different kinds take different amounts of energy to remove charges from them now where does that energy go well it goes into the kinetic energy of the electron the electron will gain kinetic energy as a result of this interaction with a Quantum of light and so finally we arrive at the following equation that the energy we’re talking about here is really the maximum amount of kinetic energy that any given electron can receive in this Collision process and that’s going to be equal to Plank’s constant time the frequency of the light that struck the electron minus the work function of the metal the minimum energy required to remove the electron from the metal this ultimately leads to the birth of the concept of the photon and implies that light has both particle like and wav like aspects that need to be taken into account now the classical description of light from Maxwell’s equations imagines that light is an electromagnetic wave with an electric field that oscillates in time and space a magnetic field that oscillates perpendicular to the electric field in time and space and that the wave travels perpendicular to both the electric and magnetic fields each wave will have an energy per unit area given by this equation this is what I said before that the intensity of the radiation is proportional to the electric field squared this is all a description but Einstein’s special relativistic description of massless phenomena which light seems to be says that the energy of a massless phenomenon is equal to its momentum times the speed of light now recall that special relativity did not tell us where the momentum itself for light comes from but that thanks to Max plunk and Albert Einstein plunk quantizing radiation and the oscillations of matter in order to explain the black body spectrum and Einstein adopting the quantization of radiation in order in order to explain the photoelectric effect and doing so perfectly then leads to the following description of light interacting with matter that the energy of a light Quantum is equal to plunks constant times the frequency or plunks constant times the speed of light divided by the wavelength of the r radiation and we see that the energy of the light is related to the frequency and that the momentum is also related to the frequency or the wavelength the origin of the numerical value of a light quantum’s energy is wavelength and frequency those tune and control the energy and momentum of a light quantum now the wav likee aspects of light like defraction and interference oscillating charges making electromagnetic waves and electromagnetic waves then also sympathetically causing charges to oscillate these are all very wav likee things that had all been very well confirmed prior to the early 1900s but the black body problem and the photoelectric effect couldn’t be solved with those wav likee aspects you needed particle like aspects of light and these phenomena began to hint that those were needed light energy comes in units that energy is defined by frequency and wavelength and light is a massless phenomenon an electron is a massive particle likee thing that can travel through space light is a massless thing that travels through space and we see from the resolution of the black body problem and the photoelectric effect that light has these Quantum discreet behaviors in the same way that particulate matter has a Quantum or discreet nature now Einstein referred to these packets of light energy as light Quant and again that comes from the Latin Quantum meaning how much now in a letter in 1926 physical chemist Gilbert Lewis uh coined the more common term the one we use today Photon implying a Quantum of light from the Greek for light now in science it’s it’s not enough to describe a phenomenon it’s important that that description have testable consequences and that there is a test that could falsify the explanation and show that it’s wrong now if your explanation survives a test it lives another day and gets to make more predictions and over time if it keeps surviving it gets adopted as an accurate description of nature perhaps even as a law of nature you can can imagine that Einstein’s explanation was not readily accepted of course and much as plank had met his own work with serious scientific skepticism Einstein’s adoption of the quantum nature of radiation to explain the photoelectric effect with all of these interesting consequences was also met with serious scientific skepticism the American physicist Robert milikin who is one of the sort of few well-known American physicists in this early part of the 1900s and his famous especially to high school chemistry students for the oil drop experiment that established the fundamental unit of electric charge although that experiment is a whole fascinating story in and of itself um Milligan did not take the claims of Einstein’s explanation about the So-Cal you know the maximum kinetic energy of an ejected electron and so forth uh very seriously he wanted to test this claim to see if it was possible to refute Einstein’s explanation of the photoelectric effect now we’re going to do a reproduction of this famous experiment by milikin in our class uh but I’ll tease the conclusion of this and and and it’s the following that milikin in 1914 after careful experimentation confirmed Einstein’s description of the photoelectric effect all stemming from the quantum hypothesis of radiation in the end the photo El electric effect paper that appeared in 1905 during this amazingly productive year of work from Albert Einstein won the day and it’s no accident therefore that Einstein went on to receive a Nobel Prize in physics in 1921 interestingly it’s for this work that Einstein received the Nobel Prize in physics not for special relativity not for general relativity but for this Niche effect in experimental physics now Einstein had extended Plank’s work to an entirely separate space of experimental effort not the black body Spectrum where plank determined the value of his constant and while he made a satisfactory explanation of that Spectrum didn’t accept the implications of his own explanation Einstein embraced those implications and then predicted all the aspects of the photo electric effect not only correctly describing what was known of the phenomena but then leading to the experiments of milikin who confirmed that description as accurate fully in its mathematical formulation this set the stage for an entirely new other perspective on nature not the theory of space and time and the speed of light and gravity the theory of the very fast but the Quantum View of matter and radiation the correct theory of the very small so to review in this lecture we have learned the following things we’ve learned how the black body radiation Spectrum was finally understood and about the possibility implied by the this resolution that energy may come in discrete units we’ve also learned about the photoelectric effect and how Einstein resolved the puzzle of the photoelectric effect by embracing the conclusion of MOX plunk work on the black body radiation Spectrum applying them to the photoelectric effect to make predictions about that phenomenon which ultimately proved to be the correct description of Nature and that all of this has set the stage for a new view of radiation and matter [Music] in this lecture we will learn about the following things we will learn about the nature of a kind of radiation called x-rays we’ll learn a little bit about the production of X-rays and and finally we’ll look at the scattering of x-rays by matter and the implications for the nature of electromagnetic radiation now X-rays were discovered serendipitously in 1895 while Wilhelm renan was experimenting with what are then known as cathode rays and which we would now simply know as electrons he was using a device that would boil electrons off a metal using a very strong electric field and he observed some distance away from the apparatus that a special phosphorescent screen was glowing even though there should be no radiation from the experiment actually reaching the screen and so he became obsessed with trying to understand this phenomenon and after careful experimentation he decided that he had isolated a new kind of radiation that was here to for unknown and using the variable for an unknown quantity in math which is usually X he coined the term x-rays to describe these now one of the things that he observed during his experiments was that if he allowed the x-rays to pass through his hand it would cast a shadow on a screen behind the hand that showed only the bones of his hand and in fact this led to him attempting to make the first what we would Now call Medical x-ray in 1895 he used the hand of his then spouse anal VI and her hand famously is the first medical x-ray ever known to have been recorded in the history of science you can see here the dark areas that look very much like the bones of the hand uh the knuckles are up here she’s clearly wearing a ring or something around her finger here and the tips of the fingers are up here the thumb is off to the side uh in for Public Presentation uh ran made a much nicer version of this picture using a different hand and a different experimental apparatus but essentially this is the birth of Medical Imaging as we think of it now noninvasive Imaging using radiation or something else to see inside the body now we now know that xrays are a kind of electromagnetic radiation they’re a very short wavelength light you can’t see them with your eyes but if you have the right instrumentation which rgin did when he serendipitously discovered them uh you can induce a signal in something that can be seen with the eyes they have wavelengths that range at their smallest between 0.01 nanm all the way up to 10 nanm now as Ren discovered they easily penetrate common lowdensity materials think cardboard skin muscle most x-rays will pass through those undeflected unstopped now if you use more dense material between U and the source of x-rays then of course what he observed was that more of the x-rays are stopped so the light regions here are places where x-rays easily made it through the dark regions are places where many fewer x-rays penetrated through the hand in order to get to the Imaging device on the other side in this case photographic film so lead bone this is more dense than skin muscle paper cardboard and so it’s more likely to stop or scatter x-rays now you can imagine that these are insanely useful not just for practical applications but for all kinds of interesting studies of the natural world and they would themselves become a key object of study and ultimately would lead the way toward understanding more about the particle like aspects of light’s behaviors now let’s talk about Arthur Holly Compton and x-ray scattering experiments as I mentioned in the previous lecture in the late 1800s and early 1900s there were not many notable physicists from the United States now that’s as compared to the then European Powerhouse of both Education and Research that was long and well established across the Atlantic Ocean now one of the physicists who became very well known in the early 1900s was Ohio born Arthur Holly Compton interestingly his PhD thesis was in part on the reflection of x-rays after his PhD he received National research Council support and was then free to travel and do research abroad and he selected to conduct work at the then famous Cavendish laboratory in Cambridge England and he did this in 1919 now there he would experiment with very short wavelength flight including X-rays and gamma rays laying the groundwork for his eventual discovery of what is now known as the Compton effect he returned to the United states in the early 1920s and became faculty at Washington University in St Louis and it was there that he observed definitively and methodically now what we refer to as the Compton effect that x-ray Quant scattered by free electrons experience a lengthening of their wavelength after the scattery and that this lengthening is a strong function of the angle at which the light is scattered so how to explain this this is a very particle like picture of an x-ray striking a free electron causing the electron to scatter and itself being scattered now in classical electromagnetism a wave would come in it would start an electron oscillating the electron would oscillate sympathetically but this shouldn’t result in a change in the wavelength of the radiation the like waves on the surface of a pond it’ll make things on on the surface start to bounce up and down but the wave itself doesn’t change wavelength uh when it when it scatters through these things on the other hand uh Compton could only explain this phenomenon by analyzing this scattering process from a more particle like Viewpoint where the X-ray quanta have energy and momentum before and after the collision with the free electron and that because the energy and momentum of the quantum is changed the wavelength is changed and he came up with a precise mathematical formula to relate all of these changes so we can hypothesize as comped did based on Einstein’s 1905 photoelectric effect work which was itself based on Max Plank’s black body Spectrum work that the X-ray incident on the electron which I labeled here I for the purposes of the coming notation uh before scattering carries a total momentum that’s given by uh E equals PC for the incident moment momentum and energy this can be related through plunk relationship to the frequency of the the radiation so H plon constant time fi the initial frequency of the radiation and if we want to get wavelength into this to consider shifts in the wavelength of the scattered light then we can convert this into HC over Lambda I where Lambda I is the initial wavelength now this allows us to write the momentum of the incident x-ray as the initial momentum is plunks constant divided by the wavelength so e is most neatly equal to H * F but P the momentum of the quantum is most neatly related to the wavelength by H divided by the wavelength Lambda the scattering process then occurs and the final scattered light Quantum carries a different momentum P final equal H over Lambda final now the initial electron State we could take as being at rest and and so it has no velocity in the initial state but the final state it involves an electron that’s now been scattered at some angle fi that we’ll write down later now it has a total speed ufal and thus it has a total momentum now I’m going to be careful here I’m not going to assume that this is necessarily a slow moving electron and in fact in reality in the Compton scattering experiments these electrons come out with a whopping great amount of momentum put them very close to the speed of light so close that it is obviously safe to use the relativistic definition of momentum that is the gamma Factor that’s a function of the speed of the ejected electron times the mass of the electron times its speed now to analyze this as a scattering process involving the Collision of a xray with a stationary electron leading to a moving electron and a scattered uh light Quantum we need to only conserve total energy and then momentum in the X and Y Direction so X is here clearly labeled as the horizontal Direction positive to the right Y is the vertical Direction and it would be positive upward vertically so let’s go ahead and do this let’s start from conservation of momentum in the horizontal or X Direction This is a closed and isolated system so the initial momentum of the system that includes the X-ray and the stationary electron must be equal to the final momentum of the system that now involves the scattered light Quantum and the moving electron so if we substitute in with the equations for the initial and final momentum of the light Quantum and we put in the component of the velocity of the electron along the horizontal axis we wind up with this equation which removing the zero because the electron in the initial state is not moving simplifies to this equation here now let me comment on a few things first of all the initial momentum of the X-ray is entirely along the xaxis but only a part of its final momentum lies along the x-axis and so that’s given by HC over Lambda F its total final momentum times the cosine of the scattering angle Theta now there’s another angle in the problem it’s the angle between the horizontal and the electron that get scattered and that’s denoted fi and because of the picture up here we’re only considering the horizontal component of the electron’s momentum and that’s given by gamma m u cosine 5 so so far nothing exciting going on it’s just breaking down the kinematics of the X-ray and the final State light Quantum and the scattered electron all along the x-axis and that’s about as far as we can go right now without knowing things like five the scattering angle of the electron we need more equations and so we’re going to turn to conserving momentum in y so let’s go go ahead and write down the vertical conservation of momentum for the same problem I’ll proceed through this relatively quickly again the initial total momentum in the y direction must be equal to the final total momentum in the y direction there is no initial momentum in the y direction the X-ray is moving entirely along the x axis the unscattered electron has no velocity so the initial state is all zeros and the final state has two pieces the positive vertical component of the scattered light Quantum and the negative vertical component of the scattered electron and so we can consolidate the zeros on the left hand side and we wind up with this equation here now we have SS instead of cosiness for the two scattering angles in the problem now we could use this to solve for fi uh or at least s of fi where it’s already looking a bit nasty we can already see that this is going to be a bit of a lift in algebra let’s see if the conservation of total energy in the system holds any comfort for us in attempting to get at a singular equation that relates the initial wavelength the final wavelength and the scattering angle of the light Quantum well we’re going to conserve total energy total initial energy must be equal to Total final energy we can plug in the total energy of the initial x-ray HC over Lambda I now remember the total energy of the unmoving electron is not zero this is special relativity mass energy is internal energy and is therefore Just Energy so we have to put in the rest mass energy of the electron the final light Quantum has an energy HC over Lambda F and the final scattered electron has a total energy given by gamma time the mass of the electron time c^ 2 this involves both kinetic and internal mass energy now we can then uh just rewrite this equation without the conservation of energy stuff on the left and we arrive at this equation here relating initial and final energies nothing’s really simplified so there’s not a lot of comfort here it’s going to be an algebraic lift but these are the pieces that Compton would have worked with and in fact did work with in order to try to understand his scattering experiments from his experiment he would have known three things the incident x-ray wavelength Lambda I the scattered light wavelength Lambda F and the angle at which light is scattered Theta so the question is of course can we use algebra possibly pages of it in order to relate these things using this hypothesis of a particle like scattering process between light quanta and an electron and could we then make a prediction for the relationship between these three things well the answer is yes and I’m going to leave the lengthy algebra to the viewer or reader of all of this stuff here but basically we’re going to eventually find by working through all of this what Compton found and that is that the predicted relationship between the final and initial wavelength and the scattering angle is given by this very Nic looking equation here in fact what we find out is that from Compton’s analysis of this process it suggests that the difference in wavelengths after and before the scatter will depend only on the scattering angle of the light and some constants of nature H the mass of the electron and the speed of light Compton ultimately confirmed that this was a correct description of of these experiments by doing his own experiments and testing this idea now there are some implications from the Compton effect which is described in this formula an undeflected x-ray that is an x-ray that goes straight through the system with an angle theta equals 0 will experience no shift in wavelength the cosine of 0 is 1 1 minus 1 is zero there is no difference between the initial and final wavelength of the X-ray that doesn’t lead to a big surprise but more interesting perhaps is that if you have a completely deflected x-ray one whose scattering angle is 180° or Pi radians that is a so-called back scatter comes straight back at the source of the x-rays it will experience the maximal possible shift in wavelength and that corresponds from a energy perspective to the largest achievable kinetic energy for the electron that’s the most kinetic energy a scattered electron is ever going to get is when you have a perfect back scatter of the uh the x-rays as a result of losing energy to the electron and coming out at this 180° scattering Angle now Compton in the course of doing his experiments did observe scattered light at Angles other than those expected from Simply scattering off the electrons and from this he determined that some of the x-rays were scattering not off of Just electrons in the atoms but entire atoms themselves that is you could rework the algebra that would lead to the compon scattering formula not by putting in the mass of a scattered electron but by putting in the mass of an entire scattered uh Atomic nucleus or atom and if you do that you’ll find that scattering at the same angle leads to a much smaller wavelength shift because the mass of an atom is much bigger than the mass of an electron and that causes the wavelength shift to get much much smaller at the same angle but nonetheless you will see scattered light with an Al an entirely different set of wavelengths albeit at a lower rate at that same angle when sometimes the x-rays scatter off of whole atoms and not just electrons what this also implied was that for light with wavelengths or frequencies at the level of xrays which does span a large space of of of wavelength ranges scattering of the light behaves more like scattering particles off of other particles like bouncing tennis balls off of bowling balls or something like that rather than waves off of particles where waves would cause sympathetic oscillations in the particles but wouldn’t change the wavelength of the original wave so this flies in the face again of the purely wave hypothesis of light and it seems that under these conditions a much better and more accurate description of the way that light behaves is as if it behaves as a large collection of quanta more than as a collection of waves now this all has deeper implications that stem from the Compton effect let’s take everything from the last few lectures together into one coherent picture all together the black body problem the photo electric effect and Compton scattering Point toward a complex set of aspects of light Behavior light isn’t just a wave and it’s not just a unit or discrete thing like a particle under some conditions light behaves exactly according to classical maxweld equation Theory waves scattering off of or otherwise interacting with and causing oscillations in matter that behavior was well established by the late 1800s electromagnetic waves really can behave like waves but under different conditions suddenly one can observe that light behaves according more accurately according to a particle description a Quantum description that light is discretized in some way not continuous like a wave so in that case it’s better described as a collection of quanta the photon so many photons all acting together and that can be thought of as particles interacting with the particles that themselves compose matter electrons whole atoms and so forth so what ultimately resulted from all of this was that there are particle like aspects of light’s behavior that tend to correspond more often to when the wavelength of the light was very short that is very high frequency whereas the wav likee aspects of light’s Behavior seemed to manifest or correspond more when the wavelength is very long that is the radiation has very low frequency somewhere in that space of wavelengths and frequencies between very long and very short there’s a transition between these sets of behavior wav like and particle like but what defines short and long that’s a very arbitrary distinction something that’s hot to one person may actually be kind of chilly to another think about uh the way that offices are heated or air conditioned uh some people find the temperature in a typical office setting perfectly fine and acceptable some people have to put a blanket over themselves to stay warm because they view it as chilly okay but what how do we Define short and long to understand when the wave and when the particle like phenomena are applicable when it turns out that the answer has to do with the dimension D or size or scale of the system with which the light interacts so the longness or shortness of of wavelength or the highness or lowness of frequency is when compared to the size of the system with which the light interacts if the wavelength of the light is much much greater than the dimensions of the system think long wavelengths that are far in excess of the size of uh of atoms for instance then it turns out that wav likee Behavior rules the atom experiences light like a wave if the wavelength is much much smaller than the dimensions of the system then the system experiences light more like being crashed into by a particle where all the momentum and energy is transferred at once particle like Behavior rules now in the middle as the wavelength becomes comparable to the size of the system things get very complicated and you have to be extremely careful and have an accurate theory in order to actually predict what’s going to happen in that case so there are some extreme cases the wavelength is much smaller than the dimensions of the system the wavelength is much longer than the dimensions of the system those are easy to handle when absolutely particle-like Behavior or absolutely wavelike Behavior manifests in the middle things get dicey and in order to describe systems which have comparable sizes to the wavelength of light for instance you need the Right theory we don’t quite have it yet now let’s talk about the sizes of things as a primer for what’s to come in our thinking about the interactions of light and matter to probe the scales of things with sizes larger than a virus and we can see from this chart over here that a virus has a size scale that’s roughly 100 nanometers okay you you can find that it’s sufficient to use visible light bacteria have uh sizes of about one micron 1,000 nanometers in size red blood cells 10,000 nanometers or about 10 microns in size hair is about 100 microns or 100,000 nanometers in size ants are 10 the 6 nanometers baseballs are about 10 to the e8th nmet we’re in the realm of the macroscopic macroscopic here meaning larger than the wavelength of visible light so this helps us to understand a little bit about why it is is that we didn’t get ourselves into trouble with large scale descriptions of motion and radiation all Newton’s laws and Maxwell’s equations when we were dealing with things that had sizes that were much um smaller than the wavelengths of light that we were using to interact with them looking at a bacterium or a red blood cell or a hair follicle with a microscope is straightforward because the wavelength of visible light is much smaller than all of those things and so it simply scatters off of them and we can resolve the sizes of those structures quite easily when you have a wavelength that’s smaller than the structure you’re looking at you can resolve the features of that structure but to probe viruses and DNA or hemoglobin or macro molecules like glucose for instance you need x-rays you need to get down to sizes that are about the level of 1 to 5 to 10 nanometers or so uh in those cases you’re going to need something like x-rays if you want to resolve the structure of glucose hemoglobin DNA which are obviously essential to understanding modern biological functioning so x-rays are your friend when you want to probe structures that are smaller than bacteria x-rays will allow you to see if you have the right instrumentation to reveal them to the eye these sorts of distance scales but if you want to probe atoms and molecules you need to really push your x-rays you need to go down to the shortest x-ray lengths about 0.01 nanometer or so the scale of atoms is at the level of .1 nanm or 10us 10 m so x-rays can be comparable to or smaller than but not by much the size of an atom and so the particle like aspects of light begin to emerge naturally at this scale it’s no surprise that the behavior of x-rays which is an electromagnetic radiation became very particle likee when we started looking at them interacting with atomic systems like the atoms in metal and the electrons in those atoms those things turn out to have scales that are roughly comparable in size uh or a little bit bigger than the kinds of x-rays that were scattering off of them and so that’s when we got our solves into trouble when it came to the theory of light and how it’s supposed to behave when it interacts with matter it’s it’s when the size of the light got to be smaller than the size of the thing that we were smashing the light into and suddenly we needed a slightly different description of light in order to understand all that now just to tease things if you wanted to probe the nucleus of an atom there we’re talking about sizes at the level of 10 -5 M or or one femtometer and for that x-rays are just too big you’re not going to resolve things of the size of a nucleus of an atom using x-rays instead you need

something with a wavelength that’s really short like gamma ray radiation or even other things that as we’ll learn turn out to have even smaller wavelengths than gamma rays let’s revisit the phenomenon of the interference of light we’ve looked at this in class in the context of the Michaelson Morley experiment what we saw in an in-class demonstration was that light that is forced to pass through a very narrow opening will defract you’ll get a pattern on a screen some distance away from the slit that the light passes through that shows light and dark spots the bright spots are where the waves have constructively interfered and their amplitudes have added up the dark spots are where the waves have been out of phase with each each other and destructively interfere black areas are places where the waves completely cancel each other out this computer simulation imagine sending in light waves toward a barrier that has two slits in it the light can defract through either slit the resulting wavefronts that come out on the far side of the barrier then interfere with each other and if we put a screen up here on the right side we could imagine Imaging this and seeing bright spots and dark spots and bright spots and dark spots and then fainter bright spots and so forth the pattern can be controlled by changing the geometry of this setup so for instance if I increase the separation between the slits and wait a few moments for the light pattern to catch up we’ll notice that as we take more data with the screen on the right the number of bright fringes has increased we now see that what was once faint on the outside is much brighter but nonetheless we have bright dark bright dark bright dark and so forth this is a wave Behavior how can we reconcile the particle nature of light and the wave nature of light in a phenomenon like this two slit light defraction instead of imagining waves of light coming into a system with two slits in the barrier let’s instead set up a situation where we can fire say photons that correspond to green light at a barrier with two slits in it one at a time if we do this we imagine sending in one Photon that Photon has to go through either the barrier on the left or the barrier on the right we don’t know which barrier it’s going to go through but we can look at the screen on the other side to see where it lands and slowly one Photon at a time as we look at the observing screen on the far side we see green dots the green dots indicate where the photon that we fired ultimately wound up on the screen one Photon at a time we’re building up an image on the far side of the screen now this is rather tedious we’d like to see if a pattern emerges in all of this so I’m going to speed this up and then when sufficient information has been received by the viewing screen on the other side I’ll comment on the pattern sufficient time has passed that we can begin to comment on the pattern we observe on the detector screen there are places where photons have clearly clumped after passing through the two slit process there are places where we find few or no photons on the detecting screen for instance those darker regions flank the bright region in the middle this is akin to the interference pattern that we saw when we were thinking about light as a wave traveling through this system and interfering with itself now single Photon by single Photon we’re building up a similar intensity pattern on the screen on the far side there are bright bands dark bands bright bands dark bands and so forth the same alternating pattern of high intensity and low intensity that we saw from the wave Behavior indeed it seems that the wave behavior is recovered in the limit of a large number of photons passing through the system this reconciles the wave and particle Behavior aspects of light in a single experiment and in fact this famous Young’s tolit experiment is one of the many ways that one can reconcile and understand these dual aspects of the existence of the phenomenon we call light in fact what seems to be true for the single Photon experiment is that we’re unable to predict with certainty where any single Photon will wind up striking the screen on the other side but the probability that a photon will strike in the middle is much higher than the prob ability that it will strike just to the right or just to the left of center and from that we can begin to build an understanding that the probability of where a photon goes on the screen seems to be somehow related to the intensity the amplitude squared of the light wave description of nature now in this lecture let’s review what we have learned we’ve learned about the nature of X-rays and we’ve seen a little bit about how to produce them by experimenting with cathode rays electron smashing into a Target we’ve then looked at the scattering of x-rays by matter and following Arthur Compton’s explanation of his scattering experiments have come to understand something about the nature of radiation with very short wavelengths from this we’ve seen the implications not only for the nature of electromagnetic radiation as having both particle like and weight like aspects under different conditions but something about the conditions themselves that trigger these different aspects of the behavior to be observed when the wavelength of the radiation is much smaller than the scale of the thing that it’s scattering off of then we see the particle like aspects of light’s Behavior emerge when the wavelength is much greater than the size of the thing off of which the radiation is scattering then we see the wavelike aspects of the r radiation emerge and in between there’s a transition a place where we lack a theory so far to actually understand how to calculate these are the foundations for what will happen next as we depart the comfortable world of radiation with its wavelike behavior and now its newly understood particle-like behavior and turn our eye from radiation to matter [Music] itself in this lecture we will learn the following things we’ll learn about the structure of the atom as it was known in the late 1800s and very early 1900s we’ll learn about how matter itself can have way aspects to Its Behavior we’ll learn about Lou de bry’s experimentally verified conjectures about the wave properties of matter and we’ll learn about how to conduct experiments that reveal the wave aspects of Matter’s Behavior to review let’s take a look at the things that describe the wave aspects and the particle aspects of electromagnetic radiation recall that the wave description of light was really the first and most formally developed part of the description of Its Behavior set and these are included in Maxwell’s equations now they describe a spatially and temporally distributed phenomenon you can boil the wave equations in Maxwell’s equations down to this set of equations describing space and time variations in electric and magnetic fields and these variations propagate at the speed of light in the material under configuration for Simplicity we can assume empty space or simply the vacuum and in that case the Solutions in empty space are the famous electromagnetic wave Solutions an oscillating electric and magnetic disturbance that travels at the speed of light perpendicular to the variations in electric and magnetic components and then of course there’s an energy per unit area of an electromagnetic wave in empty space there’s no one place where the energy is concentrated there’s more in some place and less than others and one can think about the energy density or the energy per unit area of a traveling electromagnetic wave phenomenon now the particle description of Light which emerged from evidence based on the black body radiation spectrum and the photoelectric effect these are descriptions of something that has definite energy and definite momentum at a definite location in space and time that’s what a particle is it’s a localized phenomena at a very specific place in SpaceTime whereas a wave is a spread out and distributed phenomenon that isn’t only in just one place in SpaceTime now from Albert Einstein’s work on the photoelectric effect which built upon Max plunk work with the black body Spectrum we have sort of a combined description of the particle like aspects of light’s Behavior set so for instance in special relativity we have massless phenomena whose energy and momentum are related by the speed of light eal P * C but unfortunately in special relativity we couldn’t Glimpse where the energy or the momentum of light came from we got nonsense answers from Pure special relativity however Max Plank’s work with the black body Spectrum revealed another relationship that the energy of a Quantum of light a photon is related to the frequency of the light e equal HF and the constant of proportionality is Plank’s famous constant we can plug in the relationship between wavelength and frequency for a light wave C equals F Lambda and we can get a relationship for instance between energy and frequency energy and wavelength and more interestingly between momentum and wavelength which we get by combining the energy and momentum relationships between special relativity and the black body Spectrum so we find that for a Quantum of light a localized packet like unit of light energy is related to The Wave properties of that same phenomenon by eal HF and momentum is related to The Wave Properties by P = H over Lambda let’s Model A Long wavelength interaction and a short wavelength interaction using a beaker sitting in a tank of water this identical Beaker is going to be smashed into by a wave under two different conditions on the left a long wavelength disturbance that’s much larger than the size of the beaker and on the right a short wavelength disturbance whose size is comparable to that of the beaker this will illustrate the difference between wav like phenomena and particle like phenomena the motion on the left as the wave develops in the tank will be more gradual with the beaker gently falling and then Rising whereas on the right when the short wave crashes into the beaker it’s almost as if it’s been struck by something small and fast moving look how violent the Collision on the right is compared to the one on the left we might argue the beaker on on the right has been struck by something more particle like whereas the beaker on the left which is bobbing around has been struck by something more wav likee this helps to illustrate why a wavelength phenomena short compared to the Target will exhibit particle like Behavior whereas a wavelength long compared to the Target will exhibit wav like Behavior but what if what’s so special about electromagnetic radiation why does electromagnetic radiation get to have all the fun of having particle likee and wav likee aspects to Its Behavior set what if matter electrons protons neutrons whole atoms also could exhibit wav likee behaviors they had been experienced primarily as particulate objects definite things with locations in space and time but maybe it was just because nobody had observed the wave aspects of the behavior up to a certain point such as in the early 1900s there were hints of course that something funny was going on with matter at the scale of the atomic size so for instance it had been long known since certainly the work of Anders enstrom in the in the early part of the 1800s that the emission Spectra of elements like Hydro gas or helium gas that they emitted only certain colors white light comes in a full rainbow and if you stare at White Light closely enough for instance from the Sun you’ll you’ll see that there are missing lines of color in its Spectrum but they’re hard to detect if you take a pure gas and you excite it so that it emits light it’s much easier to see that it’s only allowed to emit certain wavelength certain colors of light here I show you the hydrogen emission spectrum revealed using a optical disc a DVD or a CD ROM the many scattering surfaces for light on the surface of the disc will spread any light that strikes it out like a prism into a rainbow but we see an incomplete rainbow from hydrogen there’s a bright red line there’s a bright blue green or cyan line there’s a blue line and then there’s a fainter purple line which you can actually see part of down here reflected in the disc now this is a more classical way of laying out the emission spectrum of a gas like hydrogen’s flattening It All Out On A Plane here’s that red line the blue green line it’s very hard to see the faint dark blue line and then there’s a violet or purple line down here these are the long wavelength emissions these are the short wavelength emissions but there are big gaps in between these things and no two elements have the same spectral fingerprints excite helium excite Neon excite argon and you’ll get a very different pattern of colored lines out of each of those why why are atoms only allowed to emit certain kinds of light that was a source of curiosity in the 1800s that could not be resolved whole mathematical patterns were observed out of the relationships in wavelength or frequency between these colored lines but nobody could make sense of why these relationships existed and where they came from now to understand what’s going on with matter at size scales like that of the atom it’s very valuable to dig back a little bit into the history of the discovery and description of the atom as a real phenomenon in nature now of course thousands of years ago philosophers and mathematicians and perhaps what would now be considered Proto scientists and engineers thought deeply about matter and they argued endlessly about whether it was continuously distributed made of only a finite number of substances or Atomic in nature that is coming in small units that could be built up into the structures we experience in nature but that was a lot of argument without a lot of evidence and our modern understanding of how to understand the natural world and the scientific method reflects the reality that speculation is fine but the final Arbiter is observation of Nature and the testing of your claims the discovery of atoms as a real feature of nature or at least a potentially real feature of nature goes back to the early 1800s when chemist John Dalton discerned that not only elements have weight and that the weights are specific to each element in proportion to for instance the weight of hydrogen um but also that when you react one element with another element you’ll only get products from the reaction that completely use up the reactants if you have the right proportions of reactants for instance you might try reacting two things one to one but have an incomplete reaction react them in a ratio of 2: one and you completely eliminate all of the original reactants that went into the process that was something that Dalton characterized and it was a strong hint that the elements come in units and that those units have rules of comp combination that only allows certain proportions of them to completely react and disappear into other final products now it wouldn’t be until 1897 although speculation had preceded in the decades before this work that Joseph John or JJ Thompson would reveal the first component of what would come to be known as atams atams themselves were not completely firmly established as the correct description of nature in 1897 but Thompson found found out by experimenting on a kind of radiation known as cathode rays in his day that they actually possess of mass but they possess of a unit of mass that is about a thousand times less than that of hydrogen now this would imply that either there’s a lighter element than hydrogen or perhaps one had ripped something out of hydrogen and isolated it in the first place to be studied he observes that these cathode rays with their very tiny masses also possess of electric charge and can be made to for instance accelerate in electric Fields or bend in magnetic fields in 1905 based on the idea that this electron which composes the the cathode rays which is the identity of the COS the cathode rays is a piece of what we call atoms he proposed a model of the atom it imagine a central large positive charge with negative charges embedded in it and this was known as The Plum Pudding model because it look very much like a a British dessert known as a plum pudding where you have a whole bunch of raisins or other fruits embedded in sort of a uniform distribution of dough which is cooked up into a dessert so imagine that the raisins are the negative charges and the positive charges the dough and the negative charges are spread throughout the dough this was Thompson’s model of the atom now that may sound ludicrous and cartoonish but the beauty of science is that that’s a conjecture that can be tested for instance you might imagine trying to do experiments that verify whether or not the negative charge and the positive charge are uniformly spread out in something of the volume of an atom that’s an experiment that couldn’t necessarily be conducted at that moment in 1905 but it was certainly possible shortly thereafter now cherry-picking my way through the story I want to focus for a moment on Marie and Pierre c um now they among many other people came to understand that unstable elements or radioactive elements that emit radiation when they Decay away um results in a a new kind of radiation that had not yet up till that point been understood now they experimented on these radiations and it was finally Ernest Rutherford who classified them into the modern way that we usually talk about emission of radiation from unstable Atomic nuclei and those three classes of radiation are alpha beta and gamma for the first three letters of the Greek alphabet Alpha radiation would eventually be revealed to be a a whole hydrogen nucleus entirely ejected from a very heavy nucleus of a very heavy atom so this would be two protons and two neutrons bound together in a very stable little unit uh and it can be spat out of an unstable nucleus that spontaneously radioactively decays now Alpha radi ration is highly electrically charged it has plus two units of the elementary charge because of its two protons and that means that it can’t penetrate very far into material but it can get into material and it can dump a lot of energy along the way um as I mentioned Ernest Rutherford came up with this classification scheme he was another physicist who is considered to be one of the greatest experimentalists if not of his day perhaps even of all time uh working in conjunction with the physicist Hans Giger and Ernest Marsden he scattered Alpha radiation off metallic targets and he found out by looking at the scattering process that the plum pudding model of JJ Thompson did not describe what happened when you scattered uh alpha particles off of atomic nuclei the Thompson model would have postulated that because all the charges very spatially spread out the probability of striking any of the positive charge or any of the negative charge is extremely small and so for the most part you’d expect to find your Alpha radiation traveling through the atom lightly scattered but mostly coming out on the other side of the target but when Weatherford asked Giger and Marsden to look at what’s called back scattered alpha particles that is look for alpha particles that strike the metallic Target and then reflect almost exactly back at the original emitter of the alpha radiation they were surprised to find out that there are a significant number of alpha particles that bounce back off of the matal target as if they’re striking a huge Target of positive charge concentrated somewhere in the center of every atom and this in fact was a picture that Rutherford used to build his own model of the atom modifying JJ Thompson’s model and concentrating all the positive charge in each atom at the center of the atom this forms the first sort of planetary model of the atom as electrons orbiting a central tightly packed nucleus with a huge positive charge of course depending on the element in question but it was this picture that adequately described the back scattering process with its higher rate than expected from the Thompson model uh observed by Rutherford Giger and Marson this is now known as the Rutherford model of the atom and it would be further modified as more experiments were conducted on this system now how do we know the sizes of atoms well skipping ahead a little bit in the story of the atom you can look at the scattering of of x-rays for instance we looked at Compton scattering in a previous lecture uh but imagine scattering x-rays with slightly longer wavelengths than we would have been talking about when talking about Compton scattering here the X-ray is it turns out comparable in size to the atoms off of which it’s scattering you know with wavelengths of about .1 nanometer or so um smashing these x-rays into crystalline solids like table salt sodium chloride it was observed that specific patterns will appear in the scattered x-rays so for instance this image on the right is the very first x-ray defract made by Max von Lao Paul nipping and Walter Friedrich in 1912 now not long after Rutherford’s experiments revealed that the atom was composed of electrons with a tightly packed po positively charged nucleus now um Von Lao nipping and Friedrich noticed that there were bright spots where the x-rays tended to accumulate and dark regions where no scattered x-rays tended to be observed and this interestingly enough looked like a interference pattern that you would expect from light interfering and scattering in different ways off of a Target so using these interference patterns um and especially through the work of William Henry Bragg and William Lawrence Bragg the only Father and Son team to ever won the Nobel Prize in physics um they were able to explain the scattering of the the x-rays as being off of small objects albeit with comparable size to the x-rays in question and separations in space that were similarly comparably sized so William Henry Bragg and Lawrence Bragg did their own scattering experiments and Laurence brag in particular developed a model of the scattering process of scattering x-rays off of Regular layers of atoms in a crystall and solid that beautifully explained these patterns of light and dark that were observed at first by Von La nipping and Friedrich in 1912 and this actually led to the ability to determine the approximate size of atoms using these x-ray defraction patterns let’s take a look at the model that Lawrence Bragg developed because it will help us to understand how we can detect wave properties in general going forward let’s begin by modeling a crystal as a series of regularly arranged atoms layered in Planes we’ll come back to the separation between the planes later but they could be represented by some distance D which will appear later in this example let’s then imagine that we draw an incoming x-ray that scatters off of one particular particular atom in a plane at the top of the system now from the place where this Ray has been emitted the X-ray will strike an atom and Scatter off of it this will have a certain path length associated with it the default length that this x-ray had to travel during the scattering process we can imagine then that this x-ray came from the plane of emission shown here which makes a 90° angle to the original x-ray a second Ray emitted from very close by from the plane of emission which which also makes an angle of 90° with respect to that surface strikes another atom nearby missing the first one but hitting one in the layer below it that Ray also scatters and is detected at another point where the first x-ray is also detected photographic film or a camera or some system like that now because the second Ray did not strike the same atom as the first x-ray there’s going to be an extra bit of distance that the second x-ray has to travel before coming back to the plane where the first x-ray is also detected so we can imagine considering what that extra length is by drawing another line parallel to the line of emission the plane of emission that represents the extra distance that the X-ray would have to go that’s highlighted here in red this is the extra length that the X-ray the second one scattering off the second atom has to travel before it returns to the same location where the first x-ray also strikes a detection system on each side of the scatter off the second atom we have an extra length capital L that the X-ray had to travel and we can start doing some geometry to figure out how one relates that extra length L to the displacement D between atoms and the planes of the crystal notice that the angle between the black lines which are parallel to the plane of emission and the red lines here must also be 90° this is some geometry that you yourself could work through to verify but that Ray will always remain perpendicular to the plane of emission now the scattered x-rays will make an angle Theta with respect to the surface of the Crystal and if one works through the trigonometry and the geometry of the problem you’ll find that there is one interior angle inside the little Tri triangle whose hypotenuse is D and who each have a side of length L and and the similar angle is indicated here now we can relate the length L this is half the extra length the Ray has to travel to the distance D and the angle Theta of scattering by simply noting that in this triangle the S of theta is equal to l the opposite side divided by D the hypotenuse of the right triangle now let’s think about what’s going to happen if these two waves one scattered off of one atom on the surface of the Crystal and one scattered off of an atom in the next layer of the crystal meet at the same place on the detection screen at the same time one of these x-rays the first one for instance is a wave and it’s going to have crests and troughs just like any other electromagnetic wave now its partner x-ray that arrives at the same time will interfere constructively or destructively depending on the alignment of the second Ray with the first one let’s imagine we want to figure out what the condition is for completely constructive interference that is where the peaks of X-ray 1 line up with the peaks of X-ray 2 and the condition for that is that they be shifted relative to each other by exactly an integer number of wavelengths this is the condition for constructive interference the waves can be shifted in Distance by some distance 2 L with respect to each other but the condition is that that distance 2 L has to be an integer multiple of the x-rays wavelengths after scattering so n * Lambda so that is an integer number n * the wavelength of the X-ray Lambda meets the condition for constructive interference when n is an exact integer multiple of Lambda as I said the condition for constructive interference is that n * Lambda is some distance D and that’s the extra distance that the second x-ray has to travel and from our picture that’s twice L now we can relate this extra length L to the angle of scatter of the x-rays Theta using the trigonometric relationship derived earlier and that relationship was just that sin Theta the sign of the scattering angle equals L the side opposite that angle divided by D the hypotenuse of the triangle this allows us to solve for l in terms of D and sin Theta L is equal to D * sin Theta now plugging that into our Construction interference condition we find the following that if the second x-ray is shifted by an integer number of wavelengths with respect to the first n * Lambda then this will simply equal to 2D sin Theta constrained by the scattering requirements in the system for constructive interference and this condition this mathematical condition in order to obtain constructive interference is known as the brag condition as derived by Lawrence Bragg originally in thinking about this this x-ray scattering process so all one has to do is look at Angles where you see bright spots in the interference pattern and this will tell you given the wavelength of the x-rays what is the space separation of the planes of atoms in the crystal now in the specific case of the sodium chloride x-ray scattering that I hinted at earlier if you take regular crystals of sodium chloride and expose them to a beam of x-rays you can look to see where in scattering angle uh relative to the incident beam the bright spots and dark spots appear so for instance we have here an x-ray spectrometer the vertical axis is the number of x-rays per second that are detected and the horizontal axis is the angle with respect to the incident beam of x-rays now Theta here is the SC shattering angle with respect to say the surface of the material but this can be related via 2 Theta back to the original angle to the beam you’ll notice that there are in fact places where there are buildups of intensity of scattered x-rays so for instance just before 30° around 28° or so and just around 32° and then there’s another clump of Peaks over here uh there’s a clump just around 60 or so uh and so forth and then there’s another Clump over here there’s a very low bump and then a larger bump and you’ll notice that these bumps come with different intensities well what’s going on here is that a a copper emitter is being used to generate the x-rays and because of the properties of copper it generates two kinds of x-rays in the beam the so-called copper K Alpha line and the copper K beta line the K Alpha line has a wavelength of about .15 nanm and the the K beta line has a wavelength of about .14 nomer so they’re not exactly the same wavelength and that explains why the first bright Fringe in the X-ray has two peaks one from each of the K Alpha and K beta lines the second bright spot in the X-ray scatter has two peaks again one from the alpha and one from the beta line and so forth now if you take the Lawrence brag scattering approach and you relate the locations and angle space of bright spots constructive interference locations back to the size of the scattering uh distance between scatterers in the crystal lattice you can actually estimate the separation of the atoms or molecules that make up the crystal lattice and you find out that this comes in at about 28 nanm regardless of which of these x-ray lines you consider so we find out that the spacing of the scatterers inside a sodium chloride crystal is about the same scale as the X-ray wavelengths it’s only about a factor of two or so larger than the X-ray wavelengths that’s easy then for us to see the wave nature of the scattered x-rays emerge because they are a little bit bigger than but comparable in size to the things off of which they’re scattering uh it’s no wonder we don’t see strong Compton scattering here the particle nature of the x-rays is not in effect the wave nature of the x-rays because they’re large compared to the size of the things they’re scatter in off of is in effect uh but this is nice because it tells us roughly the scale of the size of the scattering objects and that comes in at about a fraction of a nanometer so this roughly tells us that the size of atoms or Atomic distance scales is at that level of about a fraction of an anomer now this tells us something about the sizes of atoms atoms come in at sizes around 10 Theus 10 m or so this unit is not in the system international but it’s known as the anstrom in honor of Anders anstrom the enstrom is about 10us 10 m and that roughly corresponds to the size of say a hydrogen atom or an atom that’s slightly larger than that now going back to atomic emission Spectra that is you know heating or ionizing a gas an elemental gas like hydrogen or helium or neon or something like that we get these patterns of light that come out you know it’s it’s as if only certain energies are per Ed for the electrons in an atom why would that be well in your mind you might start modeling the electron in orbit around the central nucleus of its parent atom as a string on a guitar a string on a guitar is confined at two ends uh it’s bolted down at two ends and tensioned and once you set the tension of a guitar string all the primary and secondary frequencies of its vibration are fixed and that’s how you you can tune the tension of a guitar string and get a specific note a note consists of a specific fundamental frequency and then a whole bunch of other frequencies layered on top of it with regular intervals and what determines the frequency is the length of the string and the tension of the string and that’s that basically says how many of each kind of standing wave with a certain wavelength can actually be found on a guitar string so perhaps like guitar strings confined at two ends electrons are wav likee and find themselves confined in a specific volume with only specific frequencies allowed that would certainly help explain why these patterns of light are so specific to each atom so we might draw in our mind a model of the atom as an electron confined to a volume like a spherical volume with a radius that’s about the size of an atom. one nanometer or so maybe it’s there that these wavelike properties of electrons which you could couldn’t really notice at larger scales clearly emerge and maybe that’s why Atomic Spectra have the properties that they have with these regularly spaced and in fact mathematically related colored lines this certainly would be consistent with observations of other phenomena like the black body cavity emitter where only certain vibrational frequencies of the walls of the cavity appeared to be allowed and that constrained the radiation that the cavity could emit so this isn’t totally alien the black body spectrum and atomic emission Spectra may be two aspects of the same behavior trying to tell us something about matter so if matter can be wavelike as well as particle like what is it that determines the wave properties of matter remember for light we had Maxwell’s equations they were built up from the careful study of the electric and magnetic forces and Fields and imer merged as wave equations that when solved in empty space told us that light was an electromagnetic wave an oscillatory phenomena with wav likee characteristics we have no wave equation for matter there is no first principles thing that we’ve experienced up through the end of the 1800s that tells us oh well of course there’s a wave equation for matter too so we don’t have a starting point for the wave properties of matter assuming they’re even real at all so in his 192 4 PhD thesis French physicist Louis de BR postulated postulated in the same way that Einstein postulated that the speed of light was the same for all observers that matter also has wave properties and not only that drawing from Plank’s relationship between energy and frequency for light and the relationship between momentum and wavelength that results from special relativity de Bry asserted the hypothesis that the very same facts would be true true for matter if it had wavelike properties so the energy of a piece of matter would be related to the frequency of the matter Wave by eal HF that’s a conjecture that the momentum of a piece of matter would be related to the wavelength of the corresponding matter Wave by H divided by the wavelength that’s a conjecture so how would one prove this recall Einstein made the conjecture based on the Michael morle experiment that the speed of light was the same for all observers regardless of the state of motion of the source of the light or the Observer of the light relative motion did not change the speed of light that could be tested by conducting experiments looking at the constancy of the speed of light with respect to motion now that conjecture along with the other postulate of Relativity had other predictive consequences for this description of space and time and those consequences were verified think about time dilation and the lifetime of the muon so how would one prove de bry’s conjecture well brag scattering offers the possibility to test this hypothesis we could for instance compute the matter wave properties of electrons and then we might try to find a system off of which we might scatter them and see if we can see the wave properties of electrons revealed by the scattering process all we have to do is find a scattering system whose size scale is slightly smaller than or roughly comparable to whatever the corresponding matter wavelengths of an electron would be so just as x-rays scattered from crystals allows the wave nature of x-rays to reveal to us the structure of the crystals once we know the structure of crystals themselves regular Arrangements of atoms we can then look at electron scattering and see see if it reveals any wave properties of electrons for instance interference well this is precisely what was done so consider the electron with its mass of 9.11 * 10- 31 kg now imagine accelerating it up to some momentum now we’re going to be fully relativistic here we’re going to use the correct definition of momentum because we might have to accelerate electrons to extremely high speeds to achieve the kinds of properties the wave properties we would need in order to see if those wave properties exist so we’re going to use the fully relativistic momentum equation the gamma factor of the electron times its mass times its velocity which we can set by accelerating the electron Now by De br’s postulates the momentum of an electron accelerated up to some speed U is going to be related to its matter wavelength by H over Lambda e so what momentum would we need to accelerate an electron 2 to probe the scale of a crystal whose spacing is going to be somewhere around the level of .1 nanm or so well we would ideally want to achieve an acceleration that gets our wav length down to something comparable to that about .1 NM now notice that momentum according to De bry’s postulate is inversely proportional to wavelength so if we want to get the wavelength down to something the size of 0.1 nanm we’ve got to get the momentum up high to some Target value now if you crunch the numbers on this this will require an electron momentum of about 7 * 10 -4 kg m/s that doesn’t really tell us much so for instance um if we used an accelerating electric potential difference a voltage to get our electrons up to this momentum what voltage would be needed to achieve that for an electron now I’m going to leave the math to you if you would like to play around with this but you need to make sure that you’re careful and use special relativity to answer these questions remember the relationship between um energy and the gamma Factor total energy and the gamma factor for an electron that’s written down here and remember also from special relativity that that can be related to the momentum and the rest Mass of the electron okay through this equation and keep in mind also the special relativistic definition of kinetic energy you’re going to need to combine all of these things to get the answer to the question what voltage would be needed to achieve this for an electron but it turns out that this corresponds this momentum corresponds to a gamma Factor that’s actually quite modest it’s only 1.03 that’s only a small fraction of the speed of light and that shouldn’t be hard to achieve for something as low mass as the electron that corresponds to a kinetic energy of about 2 * 10 -17 Jew um and if you remember your conversion of electron volts an electron volt is roughly 10 the19 Jew or so this isn’t many electron volts worth of of kinetic energy and so if you crunch the numbers and you relate the kinetic energy to the accelerating potential that would be required to achieve that for an electron with its one unit of Elementary charge you very quickly find out that this only requires about 150 volts that is no problem at all certainly in the days when this experiment was done uh and this experiment was done in 1927 achieving 150 volt electric potential difference for electrons was quite a trivial activity in that day so that scattering experiment was famously done by by two physicists Lester germer shown on the the right hand side of the photo and Clinton Davidson shown on the left and this is in fact um a a piece of the equipment of their scattering experiment with the electron emitter and the nickel Crystal that they used as a Target in 1927 uh to do the scattering and then they looked at the pattern of scattered electrons to see if any wave nature effects emerged and what’s the most obvious wave nature effects well if you see an interference pattern in the scattered locations of the electrons that is if you see places where there are intense locations where electrons scatter to and other dark regions where they don’t scatter to then you would have some evidence for the wave nature of electrons matter wave properties could in fact be real so just as an x-ray scattering if you scan over the scattering angle of the electrons from the Crystal and if wave properties manifest then constructive and destructive wave interference should occur at different angles for a fixed wavelength and thus a fixed momentum all right so this is an analogy to the X-ray scattering uh process of course that we looked at earlier with the brag scattering um so you could what you could do of course is you could uh set your voltage to accelerate the electrons to something specific to achieve a specific momentum for the incoming beam and then you could look at different angles of scattering relative to the beam to see if you see intense regions and less intense regions of scattering um in that case the brag scattering formula just applies uh if you want to see the nth bright Fringe of constructive interference the first the second the third and so forth then all you have to do knowing the the wavelength of the thing you’re scattering is look at a specific angle knowing the the size of the crystal the space in between the scatterers and the Crystal B and then the wavelength would simply be determined using de bry’s hypothesis using the momentum of the electron but actually instead of scanning over scattering angle it in fact when you can control very easily the momentum of the electrons then it’s actually easier to Simply vary the moment momentum of the Electron Beam and observe at a fixed angle Theta so don’t move around where you’re looking just observe at a fixed angle Theta and scan through voltage which changes the momentum of the beam and thus changes the degree of the wave properties of the beam as a function of voltage and as you scan over the voltage sometimes you’ll make the electrons have just the right wavelength to interfere totally constructively when they scatter and sometimes as you keep tuning the voltage around you’ll make them interfere spere totally destructively with each other and you’ll see no scattered electrons at that same angle Theta and this is what Davidson and germer did and here’s what they saw so this is the intensity of scattered electrons versus the square root of the voltage of their instrument and what you notice is that uh there is a place of course where there’s a bright intensity Peak and then it falls off to a minimum and then there’s another bright intensity Peak at a different voltage and then then it falls off to a minimum and so forth you see that there are these uh increases in electron intensity at a certain voltage and then you crank the voltage up a little bit more and the intensity decreases down to a minimum you keep cranking it it goes up to a maximum again we are seeing exactly what would have been predicted from brag scattering combined with the matter wave hypothesis this did not have to be this way but it turns out that matter also has wav likee properties that can be revealed under the right condition I conditions just to really drive this home in two Dimensions now scanning over scattering angle rather than fixing the scattering angle and scanning over electron momentum this is what an electron diffractogram looks like you see this pattern of bright and dark spots separated by gaps here we can very clearly see that electrons will intensely build up in the scattering process in some places and and not at all in other places with big gaps in between both vertically and horizontally there are very clearly bright spots and dark spots just like a laser beam that interferes with itself through passing through two slits for instance only waves can interfere with each other in this manner and in this case it’s because the Crystal and solids like nickel for instance off of which the electrons are scattered have structures that can accommodate an easily tuned electron momentum that yields a wavelength comparable to the size of the scattering uh system or a little bit larger and that’s easy to do with even modestly accelerated electrons on a metal Target so here’s what scattering and interference tell us about the true nature of both matter and electromagnetic radiation electromagnetic radiation already has a wave equation that describes its wave nature it comes from Maxwell’s equations so again we come back to this question well if matter can be revealed through experiment and observation to have wave properties under certain conditions then where’s the wave equation where’s the equivalent of the thing that comes from Maxwell’s equations that describes the wave properties of electrons protons neutrons whole atoms Etc where is it what is it you know electromagnetic fields and light propagating through empty space these are the solutions to Maxwell’s equations if we had an equivalent matter wave equation what will the solutions to the matter wave equation look like and these are all excellent questions and these are the questions that after these kinds of experiments had been done physicists really began to struggle with in the 1920s and into the 1930s now we’re going to get to the answer to this question very soon but we have some hints for ourselves already the solutions to the matter wave equation whatever they are whatever specific form they take for a very specific system an electron scattering off of a nickel Crystal an electron confined in a hydrogen atom whatever the solutions to the matter wave equation are going to be they’re going to be probabilistic in nature and we can already see see this revealed in the scattering intensity patterns from experiments like brag scattering the Davis and germer experiment and so forth the intensity of the scattering pattern seems to have everything to do with the probability of finding a particle at a certain location in space and time after the scattering process has occurred and that probability is controlled in some way by the original wave nature of the thing that experienced in this case the scattering phenomenon probability whatever our wave equation describes it’s going to be probabilistic in nature waves are a spread out spatial and temporal phenomenon there’s no one place where a wave is and where it is not there are many places where a wave can be and probability and the wave equation whatever it is are going to play a fundamental and deep role with one another in describing matter and radiation so let’s review in this lecture we have learned the following things we’ve learned about the structure of the atom as it was known in the late 1800s and very early 1900s cherry-picking our way through just a few scenes in the great story of the atom we’ve learned about how matter itself can have wave aspects to Its Behavior first hinted at although no one really understood this at the time by the nature of atomic Spectra and the black body Spectrum now it was Lou de Bry who conjectured that the same wave and momentum and energy descriptions that could be discerned from the black body spectrum and special relativity equally applied to matter like electrons that was a conjecture and that was experimentally verified using scattering experiments of matter off of other matter the the target had size scales that were comparable to the matter wavelength we were trying to assess and in fact tuning the beam of electrons to the right momentum to get the desired wavelength we we actually see that the wave properties manifest in the scattering experiment if electrons did not have wav likee aspects to their behavior we would not have seen the defrags that can be discerned from scattering electrons off of crystal in targets so that has also taught us how to conduct experiments both with light and with matter to reveal the wave aspects of Matter’s behavior and Compton scattering offers us a glimpse of how to reveal the particle aspects of the behavior of radiation and matter all we have to do is get the wavelength of the phenomenon to be much smaller than the size scale of the thing we’re shooting it at and the particle nature should manifest again these these ideas are going to play key roles going forward in everything we’re going to do with matter and [Music] radiation in this lecture we will learn the following things we’ll take a look at mechanical and electromagnetic wave equations to inspire our thinking about matter waves we’ll learn how to infer the nature of the wave equation for matter from an exercise involving the conservation of energy we’ll look at the meaning of the Waves described by the matter wave equation the so-called Schrodinger wave equation and finally we’ll look at the limits of absolute knowledge that are imposed by the wave nature of matter let’s take a peek at waves beginning with classical mechanics an introductory physics class would have taught you about oscillatory phenomena and a wave is just another kind of oscillatory phenomenon that can be described by time and space dependent functions so in introductory physics we learn that a Time varying oscillation along one dimension for instance a mass on the end of a spring that’s bouncing back and forth on a friction of the surface or up and down in a gravitational f field can be described as simple harmonic in nature and this allows us to write a mathematical function involving for instance the cosign of frequency and time and an offset from the amplitude being maximal at zero this is a typical equation you might see in introductory physics to describe an oscillatory phenomenon now here Omega is a special kind of frequency it’s known as the angular frequency and it’s given in terms of the period of oscillation which is a more familiar concept the period of oscillation often denoted by capital t is simply the time required for one cycle of the phenomenon to conclude the angular frequency is 2 pi divided the period and this essentially means that it’s 2 pi times the frequency of oscillation of the phenomenon angular frequency is the rate of angular displacement if we were to model the repetitive Behavior as going around a circle completing one cycle of the circle 2 pi radians as completing one cycle of the phenomenon now for all considerations here let’s set the phase angle the degree by which we would need to offset the cosine function to get the amplitude to match the initial conditions of our oscillator let’s set that phase angle to zero let’s set F to zero to simplify this equation if you then extend the phenomenon to two dimensions and imagine a long string for instance made from a bunch of tiny little masses each tiny little Mass bound to its neighbor as if by a little spring and we pluck the string that is we displace part of the string vertically then let it go and it bounces up and down and up and down the vibration of a string now we have a distortion in y That’s traveling along X in time and the solution to that problem looks something like this that the displacement in Y at any position X and at time T is is given by some initial y times the cosine of a spatial part K * X I’ll come back to K in a moment minus a temporal part Omega T which we’re already familiar with from the equation up here on a simple one-dimensional o oscillatory phenomenon now what is K well K in this context is known as the wave number and it’s defined by 2 pi the number of radians in a circle divided by the wavelength of the phenomenon so you can think of this as describing the number of cycles per unit distance in the phenomenon whereas the angular frequency is the number of cycles per unit time but these functions answer some question and if they’re the answers to a question what is the question well they are all solutions to a wave equation that is an equation that describes how changes in space relate to changes in time now the one-dimensional mechanical wave equation at least the one that tells you about vertical displacements and how they uh vary as a function of horizontal position and time is simply given by the second derivative with respect to time of the vertical amplitude Y and that’s equal to the uh constant squared times the second derivative with respect to space of the vertical displacement Y and Y of course is a function of x and t so if you try applying this wave equation to the solution on the previous slide you’ll see the following first of all the left hand side is the second derivative with respect to time of the vertical displacement y plugging in our function for the vertical displacement we would get uh this equation now the second derivative with respect to time of our description of the vertical displacement versus X and time taking one of the derivatives of this function results in us having to do the derivative twice first of the cosine function and then of the argument of the cosine function well the derivative of cosine is going to be the negative s function and the derivative of the argument is going to return a negative Omega a negative angular frequency multiplier and so we’ll be left with this Omega time the original amplitude y Sub 0 time a sign function of the original argument we have to take the time derivative of this one more time if we do that we want up with an additional factor of negative Omega out in front of the original cosine function and so at the end of this we wind up with an equation that’s just Nega Omega 2times the original function y of x and t now let’s handle the right hand side of the wave equation this is a constant term squared times the second derivative with respect to X position of the displacement y we plug in our function for y again now taking the first of the two spatial derivatives that we have here we wind up with a function that looks like this so the first derivative of cosine returns negative s and the derivative of the argument multiplied by that gives us a factor of K and so we wind up with this and we have to take the spatial derivative of this one more time and at the end of this we wind up with an equation that’s negative the constant squar time the wave number squar time the original function y of x and t Now setting these two things equal to each other as would be required by the wave equation we find out that the function y of x and t drops out of both sides of the equation leaving us with this simple relationship between the angular frequency squared the constant squared and the wave number squared and if we take the square root of all of this we see that we have Cal Omega over K and this is one of the velocities that’s present in mechanical waves the speed of the mechanical wave is given by the ratio of the angular frequency and the wave number now this is a very quick tool T of a solution to the wave equation and how you can see that it does solve the wave equation and how when you plug it in it returns a relationship between frequency squared speed squared and wave number which is related to wavelength squared a dedicated waves course would spend a lot more time on this motivating the derivation of the wave equation itself from a simple model of a vibrating string or something like that motivating how one sets up and solves that equation and then showing you what relationships emerge from Solutions under different conditions here I am merely trying to motivate some thought process about wave equations and the resulting relationships that can be derived from the application of those wave equations to their Solutions so sticking with mechanical waves for a moment let’s think a little bit about the energy that’s contained in that wave so again our model here is a mechanical Distortion of a physical medium and that Medium that I have in mind here might be a string made from many little bits of mass all hooked together as if by little Springs each with a spring constant and so forth so if we Model A String that way we can think about the string as having a total mass capital M and a total length capital L and the little bits of mass it’s made from are all equal in size and uniformly distributed along the length L and so this string has a uniform linear mass density given by the Greek letter mu which is mass / length Big M / by big l no matter what chunk of the string we look at every chunk will have the same mu because it’s a uniform distribution of mass and so we can always relate mu to the mass in that chunk and the length of that chunk if we then vibrate the string such that a given part of it at some time T in location X will have a small mass m and that mass will have a vertical velocity VY that velocity will uh oscillate transverse to its length just as the displacement oscillates transverse to its length that tiny little chunk of the string will have a length DX a differential of X and a little mass m that can be related to the length DX by the linear mass density so the Little M divided by the little DX would be mu because it’s a uniform distribution of mass so that means that m is equal to Mu DX in every place we see M we can replace it with this product and vice versa the kinetic energy of that little chunk in a moment of its motion uh as the string vibrates will be defined by its mass and its velocity at a given moment in time T So taking the classical definition of kinetic energy we’re thinking about a mechanical wave here so let’s think classically for a second we have the little bit of kinetic energy possessed of by that little bit of mass is going to be 1/2 time its mass which is Mu DX time its transverse velocity squared vy^ 2 well VY is just the Der I ative of the displacement in the Y Direction with respect to time Dy DT and we’re going to square that so we wind up if you plug in that derivative uh and and do that as we did on the previous uh page we wind up with this equation for the little bit of kinetic energy possessed of by that little bit of mass that makes up the string so this is the term for the little bit of kinetic energy possessed of by that little chunk of mass now because it’s hooked to its neighbors by springy things each mass is linked to the next by you could imagine a little spring with a spring constant um the potential energy stored in that same chunk of mass will depend on the elasticity of the string that is the stiffness of the little springs that you could imagine hold one chunk of Mass to the next so thinking of the string this way as concocted of a whole bunch of little masses M connected to their neighbors by little Springs with spring Constance Kappa then as an introductory Mechanics for oscillatory phenomena masses on a spring you can relate the angular frequency squared to the ratio of the spring constant and the mass that is to say the spring constant is related to the mass times the angular frequency squared for an oscillating Mass on the end of a spring with spring constant Kappa so the little bit of potential energy that’s stored at that location at X in that little mass m is just 1/2 time the spring constant times the displacement from equilibrium squared well that’s just going to be little chunk of potential energy held by that little bit of mass 1 12 * Kappa * the displacement squared we go ahead and substitute for Kappa with M Omega s and we can substitute for M with mu DX and then finally we can put in our equation for the displacement and that now involves the cosine and that whole thing is squared so we have kinetic energy we have potential energy let’s look at the total energy possessed of by this little bit of mass m so that little bit of mass m will have total energy de composed of kinetic and potential added together at any moment in time we have the expressions for those two things DK and du and you’ll notice that if you pull out all the multiplicative factors um you’ll be left with the same coefficients multiplying a s term squared and the same coefficients multiplying a cosine term squared so you wind up being able to pull all those multip itive factors 12 and Y and Omega squ and mu and DX out in front of a sum of a sin squ and a cosine squar and there’s a trigonometric identity that comes into play sin squ plus cosine squ is 1 and so the S and cosine functions vanish from the total energy of this little chunk and all that defines its little bit of energy that it possesses at any moment in time is that the total energy of that little chunk is constant it may be divided differently between kinetic and potential but the total energy of that chunk of mass that makes up that string that’s vibrating is constant and it’s given by this number here and again it depends on angular frequency linear mass density the length of the element the initial displacement of of any element of the string and so forth this is just the energy stored in this little piece of mass m at a location X in space and T in time now note that the total energy depends on the square of the angular frequency the presence of the Omega squ multiplier tells us something about the number of time derivatives or the product of the number of time derivatives that had been present in the original equation for total energy remember we had to square the time derivative of the wave function that solves this mechanical wave equation and that yielded an Omega squar term in all of this so you can see that there are shades of the number of derivatives uh left over as sort of vestigial elements of the energy equation for this little bit of mass m so here are the key takeaways from this look at mechanical waves the mechanical wave equation relates the second derivative with respect to space and the second derivative with respect to time you would derive its form in a dedicated class on waves but we don’t have time for that here nonetheless I want you to take away the big lessons from this now the solutions to the wave equation when acted

upon by the derivatives in the wave equation yield squares of the angular frequency Omega and the wave number K recall we had an equation relating Omega squ k^ 2 and the speed of the wave squared the energy equation for the wave or a part of the wave is sensitive to the number of time or space derivatives in the underlying equation and these manifest as multipliers like Omega squar and one might think about the presence of the squares of these quantities like Omega or k as indicative of the underlying wave equation that you needed to have solved in order to get these Solutions in the first place now let’s take a quick look at waves in electromagnetism this is the next classical wave equation that was discovered in the history of physics and it’s derived from Maxwell’s equations for electric and magnetic fields so the wave equation that results describes the propagation of oscillating electric and magnetic fields in in empty space for instance although it’s not limited to only empty space and that wave equation can be written as follows that the speed of light in empty space squared times a spatial derivative squared minus a Time derivative squared all of this acting on an electric field Vector is equal to zero well again notice that like the mechanical wave equation we’ve got second derivatives in space and second derivatives in time all acting on a solution e Vector whose form we don’t necessarily know beforehand but if we solve the equation we find out that the solutions to the oscillating electric field components look very similar to the mechanical waves in that they have a vector amplitude instead of a scalar amplitude a cosine function a spatial piece and a temporal piece of the argument of the cosine function now I should note that yes there is an identical equation for magnetic fields that can be derived from Maxwell’s equations but you can think of it as a bit redundant it describes the action of the oscillating magnetic field but can be related through the mathematics of the solution to the electric field and so if you can remember the electric field wave equation which I’m not asking you to do but will come in handy in a dedicated course on electromagnetism later um you can very quickly work out what the form of the magnetic field wave equation is and relate the electric field to the magnetic field although they are in independent directions of one another the field strengths are not independent of each other um this is an interesting problem in that electromagnetic waves are two component waves they have an oscillating electric component and an oscillating magnetic component they have two kinds of information that are stored in the wave and this is a theme that we’ll return to later when we look at matter waves now applying the wave equation written here to the solution written here similarly yields quadratic multipliers of k^ squ uh and Omega squar so for instance we find that the speed of light squared will be equal to the angular frequency squared divided by the wave number squared and this latter relationship turns out to be a direct consequence of the massless nature of light that we learn from Plank’s relationship for energy and momentum and wavelength and so forth and the special relativistic relationship between energy and momentum for massless particles uh this allows its speed to be related directly to its frequency and wavelength with no other multip multiplicative or additive factors involved so let’s revisit that the relationship between frequency and wavelength through a wave can be directly related to the energy present in the radiation Quantum the photon the photon is the particle like aspect of light’s Behavior and the electromagnetic wave is the full wave description of light’s Behavior so for example from above and from our previous look at electromagnetic waves we know that the speed of light can be related to the wavelength and the frequency of light as follows wavelength Lambda time frequency F now to get angular quantities shoehorned into this equation like Omega and K what we can do is we can multiply Lambda F by a clever number one so I’m going to multiply by 2 pi ided 2 pi if I group The 2 pi in the numerator with f i get Omega the angular frequency 2 pi F if I group The 2 pi in the denominator with Lambda I get k k K is just 2 pi over Lambda so I wind up with the equation that the speed of light is Omega / K the angular frequency divided by the wave number now recalling that Plank’s relationship for the energy and frequency of light related by plunk constant is E equals HF we can play that same game and shoehorn angular quantities in here by multiplying HF by a clever number one 2 Pi / 2 pi grouping the 2 pi in the numerator with f the frequency in the numerator and taking H and dividing it by the remaining 2 pi in the denominator we wind up with this compact equation for the energy as related to the angular frequency H bar is to denote H over 2 pi and it is known as the reduced plunks constant remember that plunk constant has a value of 6. 626 * 10- 34 so all you do is take that that divide by 2 pi 2 pi is approximately 6 and so you wind up with a number that’s about 1 * 10- 34 um it’s very convenient for all of these angular wave Concepts to carry H bar around rather than H and 2 pi so it’s very nice to Define this reduced plx constant notationally as an H with a line through the vertical part of the H now the momentum of the quantum is given by P = H over Lambda and again if you insert a clever one in that you’ll find that this is equal to H bar * K the wave number what’s nice about this is it kind of puts e and P for an electromagnetic wave on equal footing e is HR Omega p is h k these are a little easier to remember than the H over Lambda and H * F thing at least I find them more convenient once you feel more comfortable with angular frequency and wave number as angular concepts of oscillatory motion recalling that the speed of light for our electromagnetic waves is given by Omega over K that is the angular frequency divided by the wave number if we substitute in for those two quantities with their energy and momentum Expressions we recover the special relativistic relationship for light as a massless phenomenon E equals P * the speed of light so we’ve exactly recovered the Einstein energy relation for light a massless phenomenon so let’s take an overview of the wave equation In classical mechanics and electromagnetism they involve second derivatives of both space and time so you see both uh d^2 dx^ 2 and d^2 dt^ 2 in these wave equations now we could have inferred that from the results of the wave equations as their application results in squares of time and spatial frequencies so for instance Omega squares and K squar appear in equations that result from the application of the wave equations the energy equations tell us the proportionality of frequency to wavelength as well as other useful information like that and all of this kind of leaves you wondering if we have a a mechanical wave equation and an electromagnetic wave equation where’s the matter wave equation where is the evidence for that from The History of Science up to the early 1900s that’s the problem since its presence could not be inferred directly from previous measurements in the same way that Maxwell’s equations were inferred from kul’s law gauss’s law and other things like that ampers law and ultimately when composed together in the form of Maxwell’s equations led to the wave equation for light and in the same sense that considering a string is a bunch of masses Bound by Springs that are uh you know uh tugged up and down and then caused to vibrate by being stretched will lead you to the wave equation where where is the exercise that leads to the wave equation for matter and that didn’t really exist up through the 1920s the early 1920s so the question I would put to you is is it possible given other equations that we could infer from what we know its form so if we know things about particles and waves like wavelength and frequency and energy and momentum and we know the relationships between those things can we figure out the wave equation using all the information we have from Atomic Spectra the black body radiation Spectrum the photoelectric effect compon scattering and all that other stuff can we figure that out so one can Glimpse the hints of the underlying but unseen matter wave equation um sort of like seeing a shadow cast on a wall by a complex object that’s out of your line of sight but whose Shadow is projected onto a wall giving you hints about the real shape of the thing you can’t see and we can get that glimpse of the shadow of the matter wave equation by considering the conservation of energy for a particle that’s acted upon by an external Force now such a particle in classical physics would have of course a kinetic and a potential energy with specific forms depending on the the force involved in the problem now conservation of energy would require no matter what that the total energy of that particle is going to be the sum of its kinetic and potential pieces now sticking to classical physics for a moment because employing special relativity to derive the rule of matter waves involves a whole skill set of mathematics that really can’t be expected of you at this stage of a University career we’re going to stick with purely classical low velocity matter uh even small matter like electrons we’re going to have to consider moving at relatively low velocities not very close to the speed of light now obviously that doesn’t cover the full domain of phenomena of small particles like that but it’ll get us going and it will allow us to solve a great number of of problems that are actually within our grasp once we figure out the matter wave equation so using classical physics we can write the kinetic energy as 12 mv^2 and we’re going to leave the potential energy unspecified I’m not going to worry about what the force is that’s acting on this let’s just say it has a potential energy U for now and leave it at that now we have relationships for matter waves between total energy and frequency and total momentum and wavelength but we don’t have moment mum in this equation so let’s get momentum into this equation and the way we do that is we multiply yet again by a clever number one so if we insert a number one in the kinetic energy equation that is just m / M then we get an m^2 V ^2 in the numerator and MV is just momentum so we wind up with momentum classical momentum squared in the numerator divided by twice the mass of the particle plus its potential energy U so we have our kinetic term now expressed in terms of momentum and we still have our potential energy term and they’re summed together to get the total energy e so let us now inject de br’s postulates into this equation that is eal HF which is equal to H bar Omega and P equal H over Lambda which is just equal to H bar k again employing all these nice angular quantities and if we do this we now obtain the shadow cast on the wall by the matter wave equation and that is H Omega equal h^ 2 k^2 2 m + U do you see it the single power of Omega on the left side indicates to us that a shadow is cast here by a single time derivative that’s acted on some solution to the underlying wave equation we don’t see the solution and we don’t see the wave equation but we see the result of applying those two things and that is a single power of Omega on the left side the k^ squ on the right hand side implies that there’s a second derivative with respect to space in the wave equation acting on the solutions to that equation whatever they may be so we have a single time derivative and a second space derivative that result in k^2 and Omega so let’s go ahead and take that equation with our hypotheses our hunches about what the underlying wave equations form might look like and let’s try inserting those hunches into this equation above assuming that an appropriate derivative has acted on an unknown solution to yield a single Omega or a k s so if we do that if we take our hypothesis about the number of derivatives acting on some unknown function that solves the wave equation yielding this relationship we wind up with the following equation on the left H Bar times the first time derivative of an unknown solution which I’m denoting with the Greek letter SII and it’s a function of space and time we’re only considering Motion in one Dimension right now on the right hand side we have h^ 2 m time the second derivative with respect to space of that same function s of x and t and of course we have the potential energy of the the matter wave still tacked on to the right hand side over here and I’ll return to that a little bit more later so this looks promising it has all the Hallmarks of a wave equation but it’s different from Maxwell’s equations or a mechanical wave equation in one key way it has a second derivative in space but only a single or first derivative in time this will have implications for the kinds of functions that can solve such equations and the solutions to this as I’ve said are denoted by the capital Greek letter S as a function of x and t in one dimension so let’s explore solutions to this equation and as we do this we’ll find that we are missing at least one key piece of the underlying equation we’ve guessed at the form of the object casting the shadow on the wall and we may have guessed incorrectly so let’s begin by guessing the form of the solutions to our equation and then plug them in and see if we recover our energy conservation statement so to simplify matters let’s consider for now free particles that is particles free from external forces uh and that is most simply expressed by setting U to zero the particle has no potential energy associated with it we’re only considering motion at a constant velocity so that’s a fixed kinetic energy which then relates to the total energy of the particle okay well to solve our wave equation we need a kind of function that when acted on by a derivative transmutes into another version of itself so for instance in the old wave equations the mechanical wave equation the electromagnetic wave equation we had second derivatives acting on the solutions signs and cosiness were great for that because after two derivatives they return to their original selves so that’s what we did for traditional waves we used signs and cosines so let’s take a guess let’s guess that s of x and t is just one of our mechanical wave Solutions a cosine of KX minus Omega T so we’re not doing anything original here we’re just taking mechanical waves getting inspired by them and blindly applying that idea here so let’s write down our wave equation that we’ve guessed H bar time the first derivative of our solution with respect to time h 2 m * the second space derivative of our solution let’s go ahead and plug our guess at the solution in all right so we plugged our function in on both sides now and then go ahead and work out the derivatives and you should find the following conclusion that we wind up with a positive sign function on the left now multiplied by Omega we get a minus sign from the derivative of the cosine but we also get a minus sign from the derivative of its argument with respect to time so this winds up being net positive on the left side but over here the two derivatives of the cosine uh yield an overall minus sign so we have a positive s function and a negative cosine function we we can’t cancel out the signs and cosiness on either side it doesn’t recover our original energy conservation expression it doesn’t work to solve the equation the left side and right sides don’t give us what we would have expected based on where we had derived this from which was the conservation of energy and if you try just a sign function it will similarly fail so what if instead we combine s and cosine functions what if we add together SS and cosiness because when you take the first derivative of something that’s a s plus a cosine you’ll wind up with a sign and a cosine in the result and similarly with the second derivative maybe a superposition an addition of s and cosine will do the trick all right let’s go ahead and try that now I’m trying the barest simplest superposition I’m assuming that they have the same multiplicative coefficient out in front a whatever that is and otherwise it’s a cosine of the same argument and a sign of the same argument all added together when you’re playing around with solving equations whose Solutions are not known to you a priori that is with prior knowledge beforehand guesses like this will get you through the process and you should always try to start with the simplest guess and work your way up in complexity so for instance it may be these coefficients aren’t supposed to be the same but don’t start by assuming that try assuming them and then work your way up to a more General set of solutions as you get more comfortable with solving the problem so we write down our guess at the wave equation again we plug in our new choice of the solution we work through the derivatives and we’ll get the following equations now what I want to do is I want to reshuffle the term order on the left side I want to get the cosine first and then the S second so I’m just going to move these terms around without changing anything about the equation and this is the final form of the equation I get I get get a a negative cosine and a positive sign and I get a negative cosine and a negative sign over here I can’t cancel these functions out I can’t recover the energy conservation expression we started from it’s a lot closer than we were with just cosine but it’s still no good we’ve got problems with the plus and minus signs and all of this it’s a mess and what’s the real problem we keep running into here well we keep generating minus signs from the single derivative of only the cosine on the left side the derivative of the sign gives us something positive but the derivative of the cosine gives us negative s function and that’s what’s really hurting us here our goal at this point is if we’re going to figure this out we’ve got to find a way to get rid of this minus sign we keep picking up and at this point it helps to remember that there are other kinds of numbers than real numbers in the world so everything I have done up till now is predicated on the assumption that these Solutions and perhaps even the wave equation itself can only be composed from real valued numbers you know like 1.1 or 2.3 or Pi or 75.6 those are all numbers that can manifest in the real world if somebody says look I’m you know I’m going to give you -76 it means that they’re going to take $76 away from you right that has real consequences negative numbers are are real things in the world around us but there are other kinds of numbers that don’t have physical meaning in the world around us and it’s important to remember that and they fall under at least one class of these numbers is a category known as imaginary numbers and in particular it’s helpful at this point to remember some of the behavior set of the archetypal imaginary number I which effectively serves the role of being the number one in the imaginary Ary number set so let’s pause for a moment and revisit imaginary numbers which presumably you have seen in some context prior to this course let’s recall that the imaginary number I is a special kind of number one but with no physical interpretation um so I is defined by the question what squared equals -1 and the answer to that is I and I’s value would be the square root of ne1 which is nonsensical if somebody told you you know give me I dollars you would not know what to do with that because you don’t know how to calculate the square root of a negative number and then turn that into a real dollar value that you then hand that person now this number can exist in a mathematical Universe where it has plenty of self-consistent rules that don’t violate any of the axioms of mathematics that you’re uh that you’re playing with and in fact I doesn’t violate any ma axioms of mathematics at all so it’s perfectly mathematically tenable even if it’s not physically realizable around us in the world its existence mathematically has consequences though so for example uh you know going back to the question that leads to I you can take I and multiply it by itself that’s just i^ 2 if we plug in for i^ 2 we have the sare < TK of -1 * the < TK of1 and by definition that has to yield ne1 and so i^2 = 1 I is the answer to that question you know the square of what number gives me1 but you’ll notice that I S has the ability to add a stray minus sign where none would have been present before and I is the number one in the imaginary number world it is the unit on which you can build all other numbers uh integers for instance in imaginary space so the presence of extraneous minus signs when trying to solve equations using functions as we’ve been doing with our guest at the matter wave equation could actually be an indicator of something that we are trying to use real valued numbers and solutions only but maybe the problem we’re trying to solve is too complicated to only admit real numbers that it requires the ability to store additional information that real numbers alone cannot accommodate those can be accommodated by complex numbers these are numbers that contain both real and imaginary components so for instance the complex number Z is made from two real numbers X and Y but Y is multiplied by the number I and so I Y is imaginary X is real this is a combination of a real and an imaginary part this is what is known as a complex number and it stores twice as much information as a single real number because it’s got this extra component over here here and if this is reminding you a lot of vectors like a vector Z being equal to a component along the x axis X and the Y component y that’s good because a lot of the basic ideas of vectors translate into complex numbers and give us some confidence about how we can get useful information out of complex numbers so let’s explore complex solutions to this equation so let’s start by trying a guess at a complex solution it’s got a real valued part A cosine and it’s got an imaginary part a i sin so all I’ve done is I’ve added the number I to the sign part of my solution so again here’s my guess at the wave equation I plug in my solution I take my derivatives and I get the following results now again I’ve got s and cosine out of order on the left side so if I shuffle them around to get cosine first and sin 2 and try to map that onto the right I see that I still have a problem I’ve got negative I cosine and negative cosine here I’ve got a sin and I’ve got negative a I sign over here I can’t just naively cancel these things out that doesn’t really work um so I have a problem I I still can’t get this to work out if I were to try to move an i or a negative I from the left side to the right side I still wind up with a stray eye where I don’t want one and and that isn’t going to work for this problem so it’s ridiculously close to working out we’re so temptingly close to solving this problem right now but something is still missing some salt is missing from our soup here that we used to try to mimic the the the recipe for the matter wave equation so let’s see if we can get one more opportunity to think about our assumptions by moving some minus signs around to see if we can find a clue that will resolve this little puzzle puzzle so I’m starting with the last equation from the previous slide here I haven’t done anything to it yet but what I’m going to do is I’m going to pull the minus signs on the right hand side out as an overall multiplier and then multiply both sides by -1 so I’m basically moving the minus sign here to the left side of the equation so this is the net effect of doing it notice all the terms over here on the right are now positive uh the minus sign that would have been on the right is now moved to the left and overall multiplies these terms and if I go ahead and do that if I multiply it through I see that I now have I cosine here and negative s here and I have cosine here and I sign here and that has a kind of weird Rhythm to it and that’s the Clue the difference between the function on the left hand side and the function on the right hand side boils down simp simply to a missing factor of I on the left if we had originally guessed that the wave equation was I times a Time derivative of the wave function all of this would work out we would actually get the left hand side being equal to the right hand side so if we had just done this at the beginning if we traded our original guess at the left hand side of the wave equation which was all predicated on our bias for real numbers H times the first time derivative if we traded that for negative itimes the first time derivative then we’d solve our problem and let me show you that so to wit let’s revisit our guess at the form of the wave equation I’m leaving the solution completely intact that’s still the same as what I had a moment ago and what I’ve done is I’ve changed the left hand side of our guess at the wave equation to be negative I times the first time derivative of s I’ve done nothing to the right hand side plug in the solutions do the derivatives play the game again where we Shuffle the terms on the left hand side to get cosine first and sin 2 move the minus sign on the right over to the left okay and distribute that into these terms now multiply the Nega I into the function the negative I multiplied by this I gives me * -1 which is 1 positive cosine the I multipli the S function cancels out the minus sign here and leaves me with positive I * s cosine I cosine I the left and the right sides level out and the function is now the same on both sides and it can be cancelled out mutually on both sides of the equation so Bazinga as a famous TV character might say at a moment of Revelation we’ve done it we’ve cracked the underlying form of the matter wave equation let’s prove that by seeing if it returns to us the energy conservation relation that we started from we’ve only shown here that we have a wave equation that when acting on our guess at the solution to a free particle wave Returns the solution with a bunch of multiplying coefficients and so we can cancel a function out of both sides but we don’t know that we’ve recovered the conservation of energy equation we started from so let’s see if this all works out let’s review what we did starting from the total energy of a free particle which is H Omega = h^2 k^2 2m we constructed a wave equation that had the right time and space derivatives in it to return these factors of Omega and k^2 we played with oscillatory Solutions and we learned that only complex functions will satisfy an equation like this and from this we inferred a missing imag inary number from our original guess at the equation we should have had a negative I lurking on the guess of the left side of our equation the whole time in order for this thing to have viable solutions for the free particle that work out the solution guesses that we made for free particles are of this form they’re complex they have a real part and an imaginary part and plugging them into the wave equation yields this relationship H Omega equal h^2 k^2 2m which was precisely the energy conservation equation for a free particle that we began with so it’s entirely a self-consistent exercise and that should be deeply mathematically satisfying even if you’re not completely comfortable with the process by which we arrived at that but I promise you that this is not the first time assuming this is the first time you’ve ever seen this kind of strategy done for solving an unknown equation with unknown Solutions this is not the first time you’ll bust out this trick in your life if you ever have to solve hard problems so this trick is actually useful even if it feels a little clunky at first and I I hope it conveys to you the sort of incredible exercise that must have been required originally to derive this wave equation in the 1920s we’re doing this with the benefit of a century of hindsight but our predecessors did not have the benefit of this much hindsight so while I am able to look at resources and come up with ways of explaining the way of equations form to you at this level of a Physics course the people that were involved in trying to write down the matter wave equation in the 1920s did not have the benefit of this much hindsight and so they were struggling with immense difficulty in a different mathematical landscape than we are in now so putting back the potential energy piece of this this is the full one-dimensional what is known as schinger wave equation we have another function that we can tack on to the right hand side here V of x and t which is known as the potential term that function acting on the wave function sigh uh would return the potential energy of the matter wave in this case you and this is one of the most important equations in history where I’ve added back the potential energy term to complete the equation for a particle in one dimension the one-dimensional shring or wave equation is one of the most revolutionary insights into the universe in the history of our species named after Ain Schrodinger the first person in 1926 to fully determine the form of the matter wave equation now he was doing this using a whole different set of mathematical algebraic and calculus constraints that is this is not how he derived this equation but this is sufficient at this level to motivate where an equation like this might come from he was using mathematical guidelines to infer the nature of the equation that we simply don’t have the mathematical foundation for at this stage of a Physics course to do now you might look at this equation and think well this is horrible I hate time derivatives I hate space derivatives and there’s two space derivatives and one time derivatives and these functions are awful and there’s imaginary numbers and okay that might seem daunting to you and and perhaps it is but there’s actually a much more difficult part to this equation and that is that the hard part of the shring or wave equation is the solving of the equation to specific situations the real challenge of this equation is not this equation itself although it doesn’t look very pleasant I know but rather in the finding of solutions the size of x and t to this equation given different potentials V and so forth now we’ve effectively solved the free particle case and we’ll explore the solutions to that for the rest of this lecture uh but if a potential energy term is present so in other words if there are forces in a problem and you can’t ignore them and you have to include them and those result in potential energy or changes in potential energy for a particle in a problem you have to completely rework this equation and find the correct solutions that satisfy the equation equation with the correct potential term added on and that is much more difficult and that is effectively what we’ll spend the rest of this course learning how to do for different situations that map on to the physical world now I should say that solving this equation for different situations is what has allowed us to understand semiconductors it is part of what allowed the revolution in microelectronics to happen in the first place solving this equation for Atomic system leads to an understanding of where chemistry comes from and specifically doing this on a grand scale is the heart of physical chemistry solving this equation for information systems is what results in Quantum Computing and Quantum information which is a hot subject these days and is one of the many technological frontiers of our species I cannot understate how important the shringer wave equation is for all the foundations of the world we live in today but also all the potential for the great discoveries of tomorrow [Music] in this lecture we will learn the following things we’ll learn about a classical model of the atom synthesizing two semesters of introductory physics with some of the concepts that we’ve been exploring in this course we’ll learn about how to impose specifically the matterwave hypothesis on on the atom and see if we can make some useful predictions about atoms using this structure the so-called bore model of the atom let’s briefly revisit the key observational features of the hydrogen atom at a macroscopic level atoms in general when excited by an ionizing electric potential emit not a continuous rainbow of colors but rather a discrete set of color colors the so-called atomic emission spectrum shown here is the atomic emission spectrum for the hydrogen atom it has a characteristically bright red line which can be seen in the image over here on the right it’s got a blue green or cyan line which can also be clearly seen in the image over here on the right it’s got a darker blue line and a violet line and those are a little bit harder to see you can more easily see the dark blue and the Violet line in this part of the image on the right now while the atomic emission spectrum can be readily revealed by ionizing a gas in a sealed tube and then passing the light through a system that will spread it out revealing the rainbow of colors that makes up any kind of light The Mystery of the atomic spectrum goes deep each atom has a characteristic Spectrum it’s Unique to each atom that we know of in nature hydrogen is different from Helium helium is different from lithium each each of them has this pattern that’s their own we do not understand the origin of this using classical Notions of energy and momentum and matter but at last we are ready to confront this last mystery left over from the 1800s using not only what was developed in the first two semesters of physics but what we’ve been learning in this course now let’s be more numerical about the hydrogen emission spectrum we have a red line a blue green or cyan line a dark blue line and a violet line these have Associated wavelengths for the photons that carry each of these colors of light to our eye the red line for instance has a wavelength of 656 nanm the cyan or blue green line has a wavelength of 486 nanm the dark blue line has a wavelength of 434 NM and the Violet line has a wavelength of 4 110 nanm for something we’ll do later in this lecture it’s worth noting these numbers down on a piece of paper go ahead and pause the video write these four numbers down noting the colors that go with each of them and let’s save that information for a little bit later now it was Johan Balmer who worked out the mathematical relationship between these lines in 1885 these are the lines of light from the atomic emission spectrum of hydrogen that are visible to the uned human eye there are of course ultraviolet and infrared radiations from ionized hydrogen we won’t talk about those here but they’re represented in other Spectra the Balmer spectrum is the one that spans the range of light wavelengths that are visible to the human eye now balber noted that the wavelength of each of these lines is given by a simple formula a constant btimes this ratio an integer n^ 2 / by that same integer n^ 2us 2^ 2 or 4 plugging in N = 3 4 5 6 Etc Balmer was able to show that there’s a clear mathematical relationship between these colored lines here the constant B was determined to be 364.5 nanometers and all one has to do to calculate the Balmer spectrum is know this number and this formula and use the integers greater than two and you can reproduce the wavelengths present in this picture but why why is there a clear mathematical relationship between these colored lines emitted from hydrogen and why are there similar mathematical relationships between the colored lines emitted from other atoms when ionized these are deep questions Mysteries left over from the 1800s that physics in its day could not explain now let’s recall the earlier models of the atom that we explored in a previous lecture Joseph John or JJ Thompson after discovering that cathode rays were really just electrons uh and had masses that were far smaller than the lightest known element at the time hydrogen constructed his Thompson model of the atom imagining that the electrons with their negative electric charges were embedded in a larger swath of positive charge spread out in space and if one were to fire particles through such atoms they would mostly miss the electrons which are very tiny and pass almost cleanly and undeflected through this diffuse positive electric charge Ernest Rutherford and his colleagues however revealed by scattering alpha particles off of thin metal foils that in fact sometimes the alpha particles would bounce almost directly back at the apparatus that had fired them at the metal in the first place and this implied that there was some kind of densely packed small core of positive charge at the heart of each atom surrounded by orbiting electrons as if planets going around a sun this model helped explain why while many of the alpha particles would pass through the thin foil relatively undeflected occasionally one of them would suffer a collision with his densely packed positively charged nucleus of the atom and suffer an immensely disruptive Collision some of which could send the the alpha particles coming almost straight back at the source from which they had been emitted now all of this was happening in the very late 1800s with Thompson’s work and the very early 1900s with Rutherford’s work and as you’ll see as we learned more about the atom as people thought more deeply about the implication of MOX plunk adoption of the quantization of energy to explain the black body problem and Albert Einstein’s adoption of that same notion to explain the photoelectric effect models of the atom changed rapidly in response to these ideas and this then led to the ability to conduct calculations making new predictions about the behavior of atoms but also new tests of how atoms should behave themselves this was a dynamic period in physics transitioning from the classical era of the previous three centur CES into now the modern era that we would still be living in the after effects of today now let’s consider the Rutherford atom but let’s simplify the calculations and only think about an electron going around a single proton so a hydrogen like atom but only in two Dimensions the electron is bound to the proton in a circular orbit by the kulum force in the same way that the Earth would be bound to the Sun by the gravitational force in our solar system so the electron would be orbiting the proton the proton would be the central Force emitter in this problem the electron would be responding to that Force the electric force the Kum force in this case so that force would be given here by this formula the Kum force exerted on the electron by the proton would be given by a constant 1 over 4 Pi * Epsilon KN and I’ll come back to that in a bit basically this is the permitivity of free space Epsilon KN it has something to do with how electric Fields can propagate through empty space uh the product of the charges of the electron which is negative the fundamental electric charge and the proton which is positive the fundamental electric charge and that divided by the distance squared between the electron and the proton now this R would be the orbital radius of this circular orbit now I want to note here that this unit Vector R hat carries all the the directional information of this Force Now by convention R hat points from the source of the force the proton to the recipient of the force the electron it is the sign of the electron’s negative electric charge that ultimately flips the direction of that vector and has the force pointing back toward the proton that is making it an attractive Force now according to Newton’s Second Law digging back to our first semester introductory physics the sum of all four on the electron will be simply summarized by its mass times its net acceleration well what acceleration is this electron experiencing as it orbits the proton the answer is a centripetal acceleration this ultimately is a center seeking Force which results in a center seeking acceleration changing constantly the direction of the electron’s Velocity Vector so that means that the the acceleration the net acceleration of this electron has a well- defined form it’s given by v^2 / R in magnitude and its direction Center seeking will point to the center of the circular motion which again is in the direction of negative R hat that is from the electron to the proton whereas our hat is defined as being from the proton to the electron so we have all the pieces we need to build a Rutherford model of the atom in two Dimensions using these ideas of a centrally compact positive charged nucleus and orbiting electrons so let’s go ahead and do that we can set the sum of the forces which is just the kolum force equal to the masstimes the centripetal acceleration now note that there is a negative R hat on the left side negative R hat negative R hat on the right side negative R hat drops out of this entire equation and we’re left with just this equation 1 4 Pi Epsilon KN * e^2 R is equal to the mass of the the electron times the velocity of the electron all squared now I’m leaving it in this funny form because this almost instantly lets us write down the classical kinetic energy of this electron that is 12 m v ^2 if I just multiply this equation by 1/2 I immediately get the kinetic energy of this electron going in orbit around the central proton 12 * 1 4i Epsilon * e^2 R now I’m going to leave this equation unsimplified I’m going to leave this one/ half sitting out in front to ease the next step and that is Computing the total energy of this electron you can go ahead and multiply this out if you want to but uh it’s convenient for what’s going to happen next to just leave it out that way to remind us that uh doing math with this thing is going to be relatively straightforward so let me rewrite the kinetic energy of the electron here at the top of the slide now the total energy is the sum of its potential and kinetic energies for that electron at a given orbital radius r that is to say the total energy of the electron is just its kinetic energy plus its potential energy well the only Force present is the kolon force and so that means it has an electric potential energy and that electric potential energy for the electron UE will be given by its charge negative e times the electric potential of the proton V with a subscript P well the electric potential of the proton is just going to be given by 1 4 Pi Epsilon KN time the charge of the proton divided by the distance between them so we wind up with this equation for the electric potential energy of the electron 1 4 Pi Epsilon * e^2 R and that allows us to then write the total energy e as follows it’s just our kinetic energy plus the potential energy which is a negative number and you see why I left the 1/2 here it was convenient because um uh I have 1/2 times a common multiplicative thing here and just subtracting off that common multiplicative thing here and so ultimately at the end of the day I get a negative number for the total energy of the electron and that’s okay it it just means that the uh potential energy of this particular electron outmatches its kinetic energy for this particular orbital radius R so this is the final expression that I get for the total classical energy of a 2d Rutherford atom that is just considering the electron going around a stationary proton in the center let’s write that equation down we’re going to pull it up later we’re going to need it again for something that happens later in this lecture but this is about as far as I’m going to go with this for the time being now how do we get modern Concepts like the fact that the electron is actually a wave and not a point-like particle into this thing well we see the problem already with a Rutherford model it’s not going to explain the hydrogen emission spectrum right because in the Rutherford model any orbital radius R is allowed you can put any R in there and you’ll get an e out now because any total energy is allowed for the electron this cannot explain the discrete energy spectrum of electrons in a hydrogen atom we already have a sense that quantization of some sort must be present in the atom right this is the atomic emission spectrum it Bears the Fingerprints of constrained system with only specifically allowed energies determined by those constraints now the de broi postulates enter to provide the crucial missing thing the clue that will help us to understand this whole problem this is the key step that ultimately leads to quantization so for example we know that the momentum of an electron matter wave is given by plunk constant divided by the wavelength of that matter wave we can can relate the particle like properties momentum to the wav like properties wavelength using the de BR postulate now classically and because we’re doing things at low velocity we’ll revisit this as a consequence of these choices later but we’re going to start off thinking classically we can write the momentum in terms of its speed and that’s just going to be the mass of the electron times its velocity or in this case the magnitude of its velocity its speed thinking purely classically so the question we want to answer is is every wavelength of the electron possible for our electron if it is then every momentum is allowed in the atom and we’re right back where we started again if every mum momentum is allowed every speed is allowed and if every speed is allowed every radius is allowed this has consequences in a system where there are relationships between things like speed and orbital radius speed and momentum and momentum and wavelength but maybe that’s the flaw maybe the problem here is that not every wavelength through a matter wave for our electron orbiting this Central proton is actually allowed we had a discussion in class about the Schrodinger wave equation and in that discussion I drew the real part of free particle wave functions or other kinds of wave functions up on the board and I invited you the class to discuss whether or not those wave functions made physical sense I mean it’s possible to write things down mathematically that don’t make sense physically and let’s revisit that discussion I will review the Sal points here as I go through some examples and let’s apply that discussion to the electron in orbit around the proton and see what conclusions we might draw now let’s start by thinking about a circular orbit a circular orbit is quite simply one that after one period repeats again and that means if we’re thinking about the electron as a wave at a moment in time so Frozen in space at a moment in time remember it’s not only going to be a one place its wave function is spread out over space and the space over which it’s spread out is the circumference of its orbit that’s its one dimension that it’s traveling along in this problem so whatever the wave function of this electron is it had better at least obey the basic mathematical principle that when it gets back to itself at the starting point of its orbital circumference it starts all over again from exactly where it began so let’s imagine the real part of the electron matter wave and we’re just going to make one up and it might describe the electron traveling along such a circumference of an orbit of radius r at a specific time zero so we’re going to freeze this wave function which it’s of course itself is not physical but we’re going to imagine it being frozen in time at a given moment and it might look something like this um if we pick a zero point on the orbital circumference which I’ve marked here in one dimension it might be that its wave is at a local maximum the real part of its wave function might be at a local maximum there then as we move along the circumference the wave function declines and then goes negative and then it comes back and it goes to zero declining to zero again and then it goes positive now at one circumference that is at 2 pi r we see that the function I’ve chosen here nicely comes back to where it started this seems to behave itself in the sense that the wave is one continuous wave that repeats nicely in space because again we froze in time it’s possible that this wave might be waving in time but we froze in time and so in space this thing had better meet itself when it gets back to its starting point and I’ll I’ll explain physically why that needs to be in a moment so this one seems to be a reasonable candidate wave function for describing our electron in orbit around a central proton it nicely repeats itself when it reaches 2 pi r that is z the Zero Point again on its circumference uh Its Behavior is very smooth and continuous at the boundary where the orbit then repeats where that is 2 pi which Cycles back to zero again on a circle now let’s take a look at a wave function that’s also plausible mathematically but has some undesirable physical properties so what about this matter wave for the same electron at the same radius R we Frozen it in time it’s a perfectly reasonable wave function right it’s got one wavelength here it looks wavy is this a good physical wave function for describing the electron well if we look at 2i R we see that where the wave function ends up when it gets back to its beginning again is not where it started now that doesn’t per se have any mathematical negative consequences I mean this is a perfectly allowed function I can write it down uh it has something called a jump discontinuity when it gets back to its start it jumps from this value right before 2 pi r back to its starting value at 2 pi r but but this has physical consequences so because it has this jump discontinuity at 2 pi and does not cycle back to where it started uh the jump discontinuity results in a first derivative at that point that is infinite that is the derivative of this wave function with respect to space DDX at that point 0 or 2 pi r has an infinite slope now because the first derivative with respect to space in the shring or wave equation plays the role of the thing that tells you the momentum of the particle the jump discontinuity means that we have an infinite momentum point in the wave function and a place of infinite momentum is physically forbidden it just doesn’t make any physical sense if this were the case the universe would have ended long ago if things like this were possible because there’d be a particle which would contain more energy than every other particle in the universe and that would have all kinds of terrible physical consequences so this kind of wave function is physically forbidden it may be mathematically allowed but it violates physical Notions of naturalness in the world around us because the jump discontinuity has a physical consequence that is infinite momentum what about this wave function what about this matter wave for the same electron again at the same radius R does this look to you like a good wave function go ahead and pause the video and stare at it for a moment if you drew the conclusion that yeah it’s a pretty good wave function you’re you’re on the right track I mean it’s got twice the number of wavelengths in 2i r that the first one did but it comes back to where it started at 2 pi r um in fact it differs exactly by a factor of two in wavelength from the first one and in fact all waves that satisfy the relationship that their wavelength is an integer multiple of the shortest continuous and complete wave you can write down the so-called fundamental if you will all harmonics of the fundamental wave of this electron will satisfy this condition that there’s no infinite momentum Point anywhere along the physical space the circumference where it would occupy in space and in fact none in between those integer multiples will work they’ll all have the same problem that the previous example had there will be a jump discontinuity when you get to 2i R this results in a place of infinite momentum it’s unphysical so whatever the wave function that describes the electron and orbit around the proton it must satisfy this condition in order to have physical meaning an integer number n times some fundamental wavelength Lambda is going to be equal to 2 pi r the only lambdas that will work will be those that satisfy this constraint that is 2 pi r / n equals Lambda now if we utilize the de BR postulate relating momentum and wavelength then we wind up with NH over P substituting in for Lambda equal 2i R and classically remembering that P is equal to MV this puts a constraint between the radius and the speed and the integer multiple in question here NH over MV = 2i R now you’ll notice that I can move the 2 pi over to the left side and then I’ll have H over 2 pi and that allows us then to substitute with the reduced plunks constant H bar reme remember that H bar is just h / 2 pi you get this a lot when you start switching to the angular quantities angular frequency and wave number and things like that the H bar is very convenient in those contexts so let’s go ahead and just absorb the 2 pi into the definition of har bar and we’ll arrive at the following equation for the speed of the electron and its relation to the radius of the orbit that is uh M * V * r equal NH that can be Rewritten to solve for the speed of the electron ve which is NH over Mr and in preparation for relating this back to energy Concepts I’m going to go ahead and square ve so I get ve^ squar which is just this thing on the right hand side here so from the matter wave hypothesis I have a relationship between V an integer multiple of the fundamental wavelength and the radius of the orbit that determined that wavel length the first place now V and R also appear in things like kinetic energy so you can already see that we have a new constraint to throw into energy equations that will lead us to perhaps some final understanding of why it is that the atomic spectrum is discretized now before we do that I want to talk a little bit about Neil’s B’s actual postulate it’s worth noting that the way that bore attack the this problem was to postulate that there was a quantization of angular momentum in the atom that is the electron was quantized in its orbit around the proton uh now he did this in 1913 and this was about 11 years prior to De br’s work which was in 1924 so bore asserted having I guess seen that quantization worked in other problems to explain things that had previously gone unexplained he asserted that since h and H bar the reduced plunks constant have units of angular momentum that is jewles time seconds it might be in an atom that the angular momentum L is a multiple an integer multiple of H bar that those would be the only kinds of angular momenta that would be allowed in an orbital system like a 2d Rutherford atom so the angular momentum of the electron can only come in multiples of H and for a circular orbit we can relate the uh constru straint and H bar directly to the angular momentum of a particle going in a circle and that’s just P * R which classically is MVR and so this leads to the equation MVR equal nhr from B’s assertion now later on De broi would explain the reason why this works and that’s based on what we just saw on the previous slide that is if there’s a constraint on the the structure of the electron wave function requiring that the uh radius the circumference of the orbit be related to integer multiples of a fundamental wavelength of the electron then if you go back to the previous slides so go back in the lecture video you’ll see that this exact same condition resulted from the matter wave consideration so this points to the fact that uh in 1913 when bour made this assertion this is quite a bold assertion really born out of the success of the ideas of quantization in the previous a decade or so uh based on Plank’s work and Einstein’s work and so forth um this was bore being very intellectually bold and it paid off because as you’ll see this model works extremely well so finally we can take the kinetic energy equation and we can eliminate the speed of the electron in that equation in favor of the quantization condition from the matter wave hypothesis so that is here’s our kinetic energy for for the electron and orbit around a proton that can be related to the generic kinetic energy 12 mv^ 2 but we have an expression for v^2 from the quantization condition from the matter wave hypothesis and that was determined earlier to be this so if we substitute that into the equation we find out that the kinetic energy from kul’s law is equal to this kinetic energy expression taking into account the quantization of the wave function of the electron that only certain wave shapes will be allowed for a given orbital radius R and some algebra will finally lead you to this expression for the allowed radi of an atom it’s actually quite remarkable the allowed orbital radius in this 2D model is simply given by the product of an integer 1 2 3 4 Etc squared that’s n times a product of a bunch of fundamental constants of nature notice there are no variables left you have the numbers four and Pi you have the constant of nature the permitivity of free space Epsilon KN whose value is given here 8.85 * 10-2 fads per meter you have the fundamental constant Plank’s constant the reduced version squared and a reminder that H bar is 1.05 * 10us 34 Jew seconds you have the mass of the electron 9.11 * 10- 31 kg and the fundamental Electric charge 1.62 * 10 -19 Kum the only thing that can vary in here is n and n is fixed to be an integer 1 2 3 4 5 Etc so the radius of the orbit in our hydrogen like atom is simply given by an integer squared times a number so what is that number well if we stick in n equals 1 and solve we arrive at what is known as the bore radius it’s the smallest orbit allowed in the hydrogen atom because of the imposition of the matter wave hypothesis or B’s angular momentum quantization condition which turn out to be equivalent the bore radius is just this thing here and if you calculate it out it’s about half an anstrom 5.3 * 10- 11 M which is to say that the smallest a hydrogen atom can ever be with its one electron and its one proton is is about one enstrom across and as as we saw from earlier discussions in the lectures in this course an anstrom is roughly the size scale of an atom and that’s no accident it’s imposed by the matter wave nature of the electron so we find in this model of the hydrogen atom based on a classical definition of kinetic energy and momentum but with matter wave quantization imposed we suddenly find that only a fundamental orbit and its harmonics are multiples of that orbit are allowed and this begins to look a lot more like the atom that gives rise to a quantized atomic spectrum but the question is can we see that Spectrum arise from this model well to answer this question let’s again consider the total classical energy of an electron orbiting a proton at radius R but again impose the condition that n * Lambda the matter wavelength of the electron equals 2i r that led to our writing of the bore radius and then all the allowed radi of our hydrogen atom n^2 * the bore radius so we have the total classical energy of this electron I’m just repeating the expression we wrote earlier and then I’m plugging in with the expression for r n^2 a and putting in the full definition of of a here the bore radius so you wind up if you play around with it a little bit getting an equation that looks like this you get a negative a bunch of numbers and constants * 1 n^ 2 that factor in front of 1/ n^2 is – 2.19 * 1018 Jew if you go ahead and punch in all the numbers here and calculate it in electron volts this is a much more familiar number this is the famous -3.6 electron volts that is the energy of the electron in the hydrogen atom in its lowest orbit and of course this also turns out to be the energy required to fully ionize an electron out of its parent hydrogen atom if you want to free that electron completely get it away from its proton from hydrogen and put it out at Infinity you have to put in 13.6 electron volts to liberate it so the energy of an allowed orbit of integer n corresponding to radius R = n^2 * the bore radius is given by this simple equation that the energy of that orbit that specific orbit is -3.6 EV * 1 / n^2 where n is 1 2 3 4 Etc any integer this is a remarkable fact just by imposing the matter wave hypothesis on this and requiring that the wave functions be well behaved when thinking about the wave function spread over the circumference of a circular orbit we’ve immediately arrived at a quantization condition for our model of the hydrogen atom but how good is this model in order to understand how the bore model of the atom will give us the kinds of quantized energy Spectra that would result in specific specific wavelengths of light being emitted by an excited atom let’s step back and take a look at the bore model for a moment schematically the picture on the left illustrates the classical drawing of what the bore model of the atom would look like it’s very similar to the picture that I sketched earlier with a single electron orbiting a single proton here the smallest gray Circle corresponds to the Nal 1 orbit the the smallest orbit that an electron can have around the proton at the center of the hydrogen atom and that corresponds to the bore radius the Nal 2 orbit is a multiple of 2^ 2ar or 4 * the bore radius and similarly the Nal 3 orbit is going to be a multiple of 3^ s or 9 * the bore radius electrons can only orbit at these allowed radi going in a circle around the central single proton that means that if an electron is struck by for instance electromagnetic radiation a photon it could be caused to jump into a larger orbit if the electron possesses of sufficient energy to give the energy to the electron needed to transition from one orbit to the next so for instance we might imagine that the electron started in the N equals 1 orbit of the hydrogen atom was struck by a photon of sufficient energy and was able to transition to the Nal 2 orbit of the atom maybe this resulted in a complete loss a total absorption of the photon that struck it or maybe the photon was scattered losing energy and changing its wavelength gaining uh wavelength in the process becoming longer in wavelength now the image here shows the opposite of that process an electron starts in for instance the Nal 3 orbit and then spontaneously falls down into the Nal 2 2 orbit but because conservation of energy has to hold the energy difference between the Nal 3 orbit and the nals 2 orbit must go

someplace and in this case it would result in the emission of a photon so because the atom conserves energy in order to go to a wider orbit it must absorb the energy from someplace a photon with the right frequency and wavelength can do that to drop down to a lower orbit that is one characterized by a smaller integer n it must release energy and emitting a photon of a specific wavelength and frequency will do that too so let’s consider a transition that releases a photon emits a photon in the process from an orbit that’s marked by an integer n greater than M the orbit into which it falls so n is some integer m is some integer and N is greater than M the change in energy Del Delta e is going to be given by the final energy the energy of the state marked by the number M minus the energy of the initial State the orbit marked by the integer n well if we plug in the formula for the energy of any specific orbit in a hydrogen atom that’s going to give us an overall multiplicative factor of -3.6 electron volts and that’s going to be multiplied by the difference between two fractions 1 m^ 2us 1 / n^ 2 so for example for the transition from the Nal 2 orbit to the Nal 1 orbit or n = 2 m = 1 we find that Delta e is E1 minus E2 and that’s going to be given by 10.2 electron volts go ahead and work that out yourself for practice but you should find that that energy is 10.2 electron volts but but this energy must go somewhere and so this lost energy from the electron would go into the creation of a photon that then is emitted during the process and that Photon will have an energy given by H bar Omega the product of the reduced plunks constant and its angular frequency so let’s think about the photon wavelengths from electron transitions in hydrogen using this energy conservation idea and combined with the relationship between the frequency wavelength and energy of a photon we can then calculate the expected wavelengths of photons emitted from an ionized bore atom so an atom where for instance an electron starts out at infinity and comes down into one of the the low orbits or maybe starts just above and drops down to a slightly lower orbit now recall that the Balmer series The visible wavelengths of light emitted in the atomic emission spectrum of hydrogen involved an empirical relationship between wavelengths of emitted light from hydrogen given by the following formula where the integer n ranges between 3 4 5 6 and up and this integer here is fixed at two well this looks a lot like the kind of relationship you might derive from the bore model of the atom in the transition between uh say Nal 3 and n = 2 state so just to see if we’re at all matching reality let’s tabulate the energy of photons and the corresponding wavelengths of the photons that would result from transitions from the 3 to two State the 4 to 2 State the 5 to2 State and so forth and if you do that you find the following remarkable things that the wavelength of the photon emitted when the electron goes from the Nal 3 orbit to to the Nal 2 orbit is 656 NM and if that sounds familiar it should sound exactly like the red line in the Balmer series which has this wavelength if the electron instead started in the Nal 4 orbit and dropped to the Nal 2 orbit that results in a photon of wavelength 484 nanm which is blue green and is weirdly close to the blue green line in the Balmer series similarly 5 to2 results in a 432 nanm wavelength Photon that’s blue and 6 to2 results in a 409 nanm Photon that’s a violet and these are in fact to good accuracy the Balmer series lines now they differ a little bit from the numbers before and I’ll comment on that in a moment but overall the pattern is very well explained by the quantization of orbits in the atom due to the matter wave nature of the electron and thus the resulting quantization of angular momentum all Neil’s B’s conjecture in 1913 this is a remarkable fact the fact that just using a classical model of the atom combined with matter wave nature of the electron one can immediately reproduce a pattern in the world around you in this case the Balmer series of atomic emission spectrum lines this is incredible now that said if it is wise to revisit our model and compare that to what we might actually expect from a more realistic model of atoms after all atoms are not two-dimensional things they’re three-dimensional things at the very minimum and we haven’t included an extra dimension in our model we’ve only made a very good approximation to what we would expect real atoms to need to be more accurately described by but you have to admit it’s a pretty good model for what we were trying to accomplish it almost exactly repr produces the Balmer Spectrum which no previous model could do so the so-called bore Rutherford model of the atom which is what we constructed here has a few assumptions built into it the one of them is obvious it’s two-dimensional we said that outright a little bit less obvious although I hinted at it throughout this discussion is that we’ve modeled this atom as if the electron is free to move but the proton or substituting the proton with a whole nucleus with Z protons instead so 2 3 4 5 6 7 8 protons whatever you like we have the proton pinned and unmoving at the center of the atom but you know think about planets orbiting stars or planets orbiting other planets things of comparable size and mass orbiting each other one isn’t fixed while the other one goes around it rather they co-orbit a common Center and that Center is the center of mass of the system so a more accurate model would take into account the fact that the proton can also wobble in response to being tugged on by the electron via the kolon force now we’ve also obtained this model by combining a very classical picture of a planetary atom with very classical Notions of momentum kinetic energy and so forth with the matter wave idea that’s how we stitched quantum physics into this through the matter wave idea a more realistic model of the atom of course would be fully threedimensional from the start it would allow for the motion of both the electron and the proton and in fact if one puts that into this model one much more accurately captures the balm or Spectrum wavelengths they’re a little off from what’s predicted in this model but they’re almost exactly predicted by using a model where the proton can also wobble a little bit in it as it’s orbited by the electron and of course in reality we wouldn’t start from a fully classical picture we would try to exact L solve Schrodinger’s wave equation in three dimensions using as our potential acting on the wave function the kulum potential written here in full three-dimensional Glory so there’s a r Vector hidden in here that has X and Y and Z in it the truth is we are simply not ready at this stage to commit to more realism in describing the atom this was already a bit of an exhaustive exercise at the level of say coming out of introdu physics but I promise you that through the rest of this course we are going to build up a toolkit that would allow you to attack this problem in a later semester starting from the principles outlined in this course so let’s review in this lecture we have learned the following things we’ve learned about a way to develop a classical model of the atom from classical energy and force considerations we’ve done so in two dimensions we’ve then imposed quantum physics on this by sticking the matter wave hypothesis into the atom via the electron thinking about what wavelengths would be allowed for an orbit of a given radius R and then imposing that condition on the energy conservation that is derived from the classical model built in the first step this has allowed us to make predictions about the behavior of a hydrogen like atom in this model and we found that it matches remarkably well with OB observational evidence this is certainly a far more accurate description of nature than anything that has come before and this is the bore model of the atom which is a building block to a much larger picture of quantum physics the physics of the smallest things in the universe [Music] so let’s see what we have learned already from this most basic scenario a particle moving at constant speed free from external forces there is a wave equation which means that the solutions will not have definite localization okay these are waves they describe a phenomenon that’s not specifiable to any one location in space and that means they’re spread out and as a result of that and because they describe something that is oscillating we’re led to questions like what is it that’s oscillating what are the implications for measuring things like position or momentum of particles when fundamentally they’re waves and they’re not localizable to any one definite location in space at any one time uh I may not be able to know everything that I thought I could know about particles from matter waves which are really what matter is um electromagnetic waves are very iations in the strength of electric and magnetic fields and mechanical waves are variations in say the density of a medium or the displacement of a medium what’s oscillating in a matter wave we’ll come back to that now to better understand these Solutions we need to confront the mathematics of these complex functions a bit more closely and I don’t want you to be daunted by the presence of either complex numbers or complex functions they’re basically just a representation of information that becomes necessary when a problem has too much information to describe be described by only one class of number say real numbers it’s okay it just means that matter waves contain more information than real numbers alone can capture and there’s nothing scary about that all we have to do is become more comfortable with the language of complex numbers and how to get real values out of them because after all real numbers are the only things that are realized in the physical world it may be true that phenomena can be described by complex numbers and complex functions but when we make measurements of the natural world we don’t get the answer I back from it we get numbers like 5 or – 52 or 73.7 71 back from measurements those are all real valued and so regardless of the fact that the wave equation may be complex and its Solutions may be complex somehow we’ve got to get real numbers and only real numbers out of these things and to really understand that we need to take a look at complex numbers and a little bit of the algebra related to complex numbers but basically complex numbers just double the available amount of information you can store in a single number that’s all they do so our working solution to the free particle wave equation is of this form we’ve seen it a bunch of times now and it looks weirdly similar to that representative complex number Z I showed you earlier X+ i y it’s a complex structure with a real part which we could represent by X and an imaginary part which we could represent by y it looks very similar to a a simple complex number but as I’ve said observations of the natural world are conventionally Des described by real numbers not imaginary ones now that’s okay I mean we’ve already kind of hinted at the fact that complex numbers look a lot like vectors and we’re used to dealing with vectors with an X component and a y component and from those we’re comfortable summarizing the information content of a vector using the concept of its length or its magnitude a single real number you know for instance you might have a three-dimensional velocity with a VX a VY and a VZ component and that’s all very complicated but you’re very comfortable going look the speed of the particle is V where V is the square root of vx^ 2 + V y^2 + vz^ 2 a single valued real number that summarizes the overall thrust of the Velocity Vector so that’s not scary at all that’s something you’ve been doing since beginning introductory physics the question here is how does one get a single real number out of a complex one how do we get the measurable out of the complex function or number well you might just try you know your old friend the the square right Square the complex number and see if that gives you a real value but unfortunately it gives you a complex polinomial you wind up with a real number x^2 and a real number negative y^2 but a complex piece 2 ixy that’s the sum of the Cross terms of this square if only we could get rid of that cross term we’d be home free we would actually recover something that looks weirdly like the Pythagorean theorem with an X squ and A Y squ term this is almost a hypotenuse squared but it’s not real valued so this may be the h hypotenuse in some space but it’s not the hypotenuse in the real number space of measurement okay so that won’t work now instead to get a real number you need to do something like this and this is part of what defines the algebra of complex numbers you’re going to take Z and you’re going to multiply it by a special version of itself known as Zar uh this is just x + i y the original complex number times x minus i y and you’ll notice that when you do that distributed multiplication out and add all the terms together you wind up with x^2 plus y^2 and no cross terms well this looks weirdly like the Pythagorean theorem you’ve got an X component squared you’ve got a y component squared and this is somehow related to a sort of square of the complex number although this funny thing Zar is required so while that yields something more consistent with for instance your experience with the Pythagorean theorem about the length of a vector um but it does it with a complex number with real and imaginary components what is this thing Zar that we’ve employed to get away with this and the answer is that Zar is what is known as the complex conjugate of the complex number all you do to take the complex conjugate of a complex number is take all numbers I inside the number and replace them with negative I that’s it you’re going to send I to minus I you’re going to flip the sign of all the I’s and that is all Zar represents now to keep this kind of consistent with our instincts about vectors and lengths and magnitudes and things like that we have a shorthand notation for Z * Z Star Z time its own complex conjugate to indicate that it is the square of the real length the thing we would really measure as a consequence in nature if we described a problem using complex numbers and that is denoted by the magnitude or absolute value bars of Z all squared so the magnitude of z^ squ is defined as ZZ star so if you see this notation absolute value or magnitude of z^ squared in complex space that denotes the product of Z with its complex conjugate Z star that’s how you get the real valued length of a complex number or a complex function now another interesting thing about the free particle Solutions is that uh one can simp simplify the notation that we’ve been using to carry around these free particle Solutions and and that is the language of signs and cosiness and exponential functions so for instance um it’s really clunky to have to keep writing out these signs and cosiness in our free particle wave function solution to the wave equation it would be nice if we could compactify this notation somehow and Mathematics does offer us a more compact representation of the same information and will also give us some practice with imaginary numbers like like I um ultimately we will be able to summarize the free particle Solutions as a single exponential function rather than a sum of ss and cosiness to get there let’s consider a tailor expansion of the S function sin of X so the S of X tailor expanded into a series of terms becomes x – x Cub over 3 factorial + x 5 over 5 factorial Etc similarly the cosine function can be tailor expanded into the following 1 – x^2 2 factorial + x 4 4 factorial Etc notice that the S involves only the odd powers of X and the cosine involves only the even powers of X so x to the 0 is 1 x^2 X4 and so forth and the sums all have alternating pluses and minuses that are used to combine the terms together now recall that the tailor expansion of the exponential function e to the X looks like the following if you tailor expand e to the X you wind up with 1 + x + x^2 / 2 factorial Etc so if you stare at these three things for a second you’re dangerously close to being able to find some combination of s and cosine that when added together yields e to X but it’s not going to be real valued because the sign and cosine expansions have alternating plus and minus signs in front of their terms whereas the e to the X expansion is all sums and so we see a problem here we would like to use e to the X to represent some combination of s and cosine of x but we can’t do that because we have these stray minus signs on alternating terms that complicate our ability to use only real numbers to do this trick to make sign and cosine combine to get e to X well again leaving that expansion of e to the X up here let’s go back and revisit a little bit the use of the imaginary number I and the implications it might have for combining sign and cosine so note that while the expansion of e to the X involves the sum of a bunch of power of X and the S and cosine expansions have alternating sums and subtractions we might use this rule that when you see stray minus signs that they might be indicative of products of the imaginary number I we can crack the puzzle so let’s think creatively for a moment and let’s recall that i^2 = -1 and that allows us to then rewrite terms like x^2 which appears in the expansion of the cosine function as i^2 x^2 or in other words IX all sared so it’s as if we replaced the argument of the cosine function with uh I times the argument that we started with now in the S expansion we have odd numbered powers of of of X like X cubed for instance and uh that could be Rewritten as i^2 X Cub but that’s not very satisfying we have different powers of I and X in this but let’s keep in mind that if we have a term that looks instead like Nega IX cubed that can be Rewritten and you can practice this for yourself as I cubed x cubed which is just IX all cubed so with those things in mind let’s recall our free particle Solutions are of the form a * the cosine of an argument X plus I * the S of an argument X well if we stare at that for a second and we plug in the tailor expansions of cosine and S we would get this that we have a times for instance just keeping the first two terms in the tailor expans expansion 1 – x^2 / 2 factorial and we’re going to add to that I a time this expansion of sign keeping only the first two terms x – x Cub over 3 factorial now if we distribute the imaginary number I into the parentheses on the right hand side of this uh sum we can start employing the identities and relationships that I wrote up here so for instance iix cubed is just iix all cubed and x^2 is just IX all SAR so for instance I wind up with terms like this I have IX here which is fine we can leave that alone I have IX cubed and that can be replaced with positive IX all cubed and that’s done here now for the cosine I have 1 – x^2 / 2 factorial well x^2 can be replaced with IX all SAR and you’ll notice what’s happening we’re eating up the minus signs in algebra involving the number I so we wind up with a positive sum of these terms 1 + IX plus IX all 2 over 2 factorial plus IX all cubed over 3 factorial Etc if we were keeping more terms in the tailor expansion this thing here can simply be Rewritten as a * e to the iix the argument of the cosine and S was X but combining Li this way with a multiplicative I in front of the sign term we get to rewrite that sum as a e to the I * X the original argument of the S and cosine function so we’ve traded a real valued function for a complex function but it’s a much more compact notation than what we had before and this allows us to rewrite the free particle Solutions in this more compact form as a * e to the I * the quantity KX – Omega T and this is a little bit easier iier to carry around on a piece of paper than the sums of ss and cosin with the imaginary number I in only one of the two terms now what is the magnitude of our free particle solution and let’s keep in mind that we don’t know if the constant out in front of the function a is real or complex so let’s try to calculate the magnitude squared of the wave function of the free particle uh let’s do that so we’re trying to calculate the absolute value of s^ squ and remember in a complex space of functions or numbers that’s defined as s times its complex conjugate s star well what is that well s is just a * e to the I KX minus Omega T the complex conjugative s would involve changing I to negative I everywhere we see it but we don’t know if there’s an I hiding inside of the prefactor a that multiplies the exponential function so to be very careful about this in case the a is also a complex number we’re going to replace a with a star and I with negative I up here and that’s about as far as we can go with this if we now group terms together in the multiplication we have a * a star we have e i KX – Omega T and grouping the exponents together we have then I * KX Omega T these exponents completely cancel each other out to zero and we’re left with a term that’s just e to the 0 e to the 0 is 1 so this then simplifies to a * a star or just the magnitude of a s so the measure of the wave particle function for a free particle is just a real number the magnitude of a^ s but what is it that we’ve just evaluated what is this function that solves the wave equation and what is the meaning of its length these are the the questions that really racked people’s brains in the 1920s and 1930s this was a real intellectual struggle in confronting the wave nature of matter so one is forced to interpret these functions and their meaning there is no easy answer from first principles in nature about what the wave function is because it’s a complex function you don’t actually have any physical meaning to its real and imaginary Parts it’s only the magnitude of the wave function squared that has any physical meaning and so you have to lay an interpretation down as to what you think the underlying wave function is and what is waving it’s not energy because energy is a real thing it’s something else and I have to tell you that in the history of physics and you may have seen this in popular videos on quantum mechanics which often are Rife with misunderstandings of the underlying math and subject material it’s this contest of intellectual ideas that has caused the most most hand rubbing and consternation and some of the most bitter disagreements and strong opinions in the history of Science and it’s all been over a function whose Direct Value has no physical meaning because it’s based in part on imaginary numbers which themselves have no physical interpretation it’s only the real valued magnitude of the complex function or the complex numbers that have any physical meaning it’s not those numbers themselves it’s only the measure of their overall information content that has meaning in the physical world now the most practical interpretation one which has also been met with the most experimental success since Irvin Schrodinger first published his wave equation is that of a probabilistic meaning to the square of the wave function that is to say this thing the magnitude of s squared uh this amplitude squared of the wave function is interpreted as representing a probability per unit distance per unit time in one dimension in two Dimensions it’s per unit area and in three its per unit volume now to obtain raw probabilities one has to specify the exact conditions under which the free particle has been prepared for instance where was it starting from exactly and what was its momentum and things like that and then you can answer questions such as given that this is a matter wave and it’s not localized once it’s released to any one place in space what’s the probability of finding this particle between say 1 cm and 2 cm from the point of origin or what’s the probability of finding the particle a distance of 3 cm from the point of origin 1 second after it starts its Journey these are questions you can try to answer in the framework of the matter wave equation the Shing or wave equation and all the math that goes along with it we don’t have that framework available we’re going to develop that framework going forward and try to get answers to questions like this all right so that’s our goal we’re going to conclude clude our discussion of the implications of the wave nature of matter in this lecture and later lectures we’ll begin to think about specific problem statements and then how we use the shringer wave equation to attack those problem statements and interrogate the solutions to get answers that can be measured in a laboratory experiment the wave function itself is not directly accessible but its amplitude squared in different situations has physical consequences for measurement now that said because we’re mathematical beings that can imagine things that are not physically realizable in the world around us we can use some math and computer aids to try to visualize the wave function of our matter particle that’s free from external forces but to do this we have to concoct a space of the imaginary value of the wave function and the real value value of the wave function now these are not physical axes in space they don’t have physical extent remember that this is an oscillating probability probability itself is not physical but the probabilities of outcomes are physical and so it’s you have to be very careful to separate your visualization of the wave function from physical meaning which is only derivable from the square of the wave function the complex conjugate times the original wave function nonetheless because we are mathematical beings and we can think abstractly let’s attempt to visualize what the wave function of a free particle would look like without specifying how it was prepared uh in that case then it’s the solution that we’ve written down already and we can imagine thinking about the uh amplitude of the wave function along its imaginary axis and along its real axis so along its imaginary axis it’s a sign function whose amplitude starts out at zero goes to a maximum plunges to a minimum and returns to zero after one cycle and along the real valued axis of the wave function it starts off at maximum amplitude eventually goes through zero to a minimum back through zero to a maximum after one cycle of the matter wave and note that the maximum of the matter wave in the real value part of the wave function is achieved at the same location as the zero point of the imaginary part of the wave function which is what you would expect from a cosine and a s function combined together now of course if we construct this in 3D space with our imaginary axis our real axis and then the spatial location and physical space of the particle we wind up with a helical structure a helical surface that winds through imaginary and real space uh keeping in mind that we’re talking about the imaginary and real components of the wave function but at all points in space as we’ve seen the amplitude squared of this is a constant valued number that doesn’t depend on space and time and and so whatever this wave function is doing varying in its real and imaginary Parts in physical space it represents a constant probability density everywhere in space in time so there’s nothing waving in physical space in IM in the space of the wave function you have oscillation and that oscillation is related to the probability of finding the particle at that point in space at that moment in time but in physical space all you have is the magnitude squar of the wave function that’s the only physical thing that manifests in the measurable world now to close out this lecture let’s take a look at what it means to try to measure both the position and the momentum of a matter wave representation of a particle so here’s a real valued part of the wave function of a matter Wave It’s the cosine it starts at one goes to negative 1 returns to one after one cycle and you see I’ve got two two wavelengths represented in this picture I’ve ignored the complex part but it’s also waving at the same time we’ve just looked we’re looking now just at the physical position of the particle versus the value of the real component of the wave function the imaginary component of this wave also has an important role in what happens with the physical reality of the particle but it’s not shown here I just want to concentrate your energy now on thinking about what it means to measure momentum and position for a wave or at least a particle described as a wave now measuring the position of a free particle boils down to determining where it is along the x-axis so for instance I might do that by zooming in more and more on this wave and saying okay I’m localizing the particle more and more and more by spotting the little chunk of its wave function in the real valued component located at that point in space but measuring the momentum of the same particle boils down to a different observation measuring the momentum of the particle is related ated to determining the second derivative of this wave with respect to space that is determining the curvature of this wave that’s what the second derivative with respect to space tells you it tells you about the spatial curvature of the real part or the imaginary part of the wave function and it’s that curvature the degree to which the wave bends to move toward the next part of its cycle that determines momentum now it’s very easy to determine the momentum in this picture we clearly have two wavelengths we could sit down and easily determine from the information on this page uh what the wavelength of this wave is all right but we might be a little less certain about where it is because there’s a couple of cycles of its real valued part of its wave function here so maybe it’s here or maybe it’s here or maybe it’s here all right so knowing the momentum really well might preclude knowing the position really well but what if we really localize this particle to one specific place place in in position space all right so what we want to do is try to locate the particle more and more precisely by zooming in on the wave function to really localize the phenomenon to one narrow region of space and this is equivalent to identifying where it is in a range X and X Plus Delta X and then sending Delta X more and more toward zero to zoom way way way way in on a narrow slice of the wave all right but as we’ll see it’s going to become hard harder and harder to establish the curvature of the wave as we do this and thus the momentum of the wave is going to slip from our grasp now to help you with this exercise what I want you to do is really stare at the wave in this region right here where I’m indicating with the with the mouse cursor okay so really stare at the wave here right now you can clearly see that there’s well-defined curvature you could easily and readily determine the wavelength of this phenomenon how about now can you easily determine the wavelength to this phenomenon I’ve zoomed in localizing more in space where I want to see where the particle is but in doing so I’ve traded a lot of the curvature away in order to do that it’s it’s getting harder to determine the wavelength of this wave but you could still maybe do it you’ve got a peak over here and you can see how it’s declining there’s lots of curvature to determine the the momentum of this wave but how about now I’ve zoomed in even more stare at that are you confident you could determine the curv of that wave and you may be remembering the old wave but as you continue to stare can you determine the curvature of the wave well I messed with you a little bit while you were staring at the wave while I was daring you to think about the curvature of that line I did one more change to the wave I’m still zoomed way in on it but I changed the wavelength by 10% did you notice did you notice that the wavelength changed from the previous zoom in to the zoom in you’re looking at now an astute Observer might have noticed while they were staring at it that the grid behind here uh changed when I did that and that corresponds to a change in where I was zoomed in on the wave but the starting value and the ending value of the wave in this picture didn’t change the heights of the Waves where they enter the picture and exit the picture were concocted identically giving you the impression that you were confident that the wavelength was the same as the wave from before but it’s not I changed the wavelength by 10% but presented you with a similar Zoom in region and this is meant to confuse you on purpose to show you that the more you close in on the wave function the harder and harder and harder it’s going to be to determine the curvature of the wave is this line straight is it bending gently how much is it bending you don’t have infinite resolution available to you in the universe you’re going to hit a limit at some point and it’s going to get extremely hard to determine if this is a straight line or not a straight line and if it’s not a straight line you’re going to struggle with determining exactly what its radius of curvature is and that struggle is reflected in a loss of control over your knowledge of the momentum of the particle knowing the position too well comes at the cost of knowing the momentum so let me repeat that statement one more time when you’re dealing with matter waves knowing the position very well comes at the cost of knowing the momentum with any Precision knowing the momentum very well comes at the cost of knowing the position with any Precision that I reflected in my ear statement about being zoomed out looking at many cycles of the wave you’re very confident when you’re zoomed out that you know the wavelength of this phenomenon but because there are many places where the particle is likely to be and less likely to be represented by the changing amplitude of the wave in real space you’re getting kind of confused about where it might actually be is it more at one of the Maxima or more at the other maximum or more at the third maximum or the fourth maximum or the fifth maximum gaining confidence in momentum comes comes at the cost of confidence and precision and it was the physicist verer Heisenberg who worked out the mathematics of this particular issue in 1927 now the real way to do this of course is to take the wave equation and to work through the forier transform which tells you something about the information content of the wave in position and frequency or momentum space that’s a little above the ability of a course like this to work out although you are welcome to look into it on your own if you’re comfortable with uh integrals and derivatives at a high level at least at the level of say Cal 2 and Cal 3 um Heisenberg codified the relationship between the certainty or uncertainty of our knowledge in momentum and the uncertainty of our knowledge in position in what is known as the Heisenberg uncertainty principle and it’s a very definitive statement albeit an inequality it says that the uncertainty in the knowledge of momentum Delta P times the uncertainty in the knowledge of position Delta X must always be greater than or equal to H the reduced plunks constant divided two why is it that we don’t worry about knowing how fast our car is moving while also knowing its position on the road we don’t freak out about that like if we’re going to stare at the speedometer for a moment we’re suddenly going to look up and realize we’re in New York City whereas we were in Dallas at the beginning of our glance down at the speedometer that doesn’t happen in the real world you don’t increase your confidence in your current velocity and thus your current momentum and then suddenly look up and realize you’re on the Moon I mean this is essentially what we’re talking about here with tiny matter waves right is that once you become very confident you know where the particle is you suddenly lose all confidence about its momentum and vice versa well it’s no wonder we didn’t notice it h bar over2 is a number that is approximately 10us 35 Jew seconds that’s an insanely small number it’s no wonder we didn’t notice this before and that it would only manifest at the scale of things tiny like atoms or electrons or the nucleus of the atom or things like that but this statement holds for matter waves no matter what situation you’re in you cannot know the position and the momentum at the same time with infinite precision and you can see that if you did try to know one of them with infinite Precision that is Delta X exactly equal to zero so you want to know exactly the position of a matter wave so you specify an experiment that lets you get infinite Precision no uncertainty on the position you completely lose control of the momentum the uncertainty on the momentum blows up to Infinity in order to hold this as a constant that’s the only way to satisfy this inequality is if Delta P blows up to Infinity as Delta X goes to zero this is a limit imposed by the wave nature of matter it’s unavoidable you cannot know this pair of variables X and P with any simultaneously perfect Precision now now of course the why of this is buried deeply in things like the fora transform and in the algebra of matrices that is collections of numbers in multiple Dimensions which is another form of language that can be used to derive quantum mechanics which is where we are essentially at now that’s above the pay grade of this particular class but I just want to say that because you are going to encounter quantum mechanics again in a dedicated higher level course than this one and I want you to understand that I’m having to wave my hands quite a bit at this level in order to motivate this nonetheless you will have a second crack at this where you’ll begin to see the wise of all of this where is this coming from why H over two uh why this particular product of momentum and position are there other products of things that similarly in pairs are uncertain when you know one you don’t know the other and and vice versa these are excellent questions and I don’t expect you to be satisfied with this right now but this is where we can get in a course at this level level after two semesters of introductory physics so let’s review what we have learned in this lecture we’ve learned about mechanical and electromagnetic wave equations and from that we’ve learned how to infer the nature of the wave equation for matter and this has given us some ability to get at the meaning of the Waves described by the matter wave equation albeit by interpreting what’s going on based on our experience with the natural world the wave equation involves complex numbers and the solution to the wave equations involve complex functions we have to get real numbers out of this thing if we want to map it onto the real world and the only way to do this is for instance to calculate the amplitude squar of the wave function in doing that however we lose any ability to understand or map the physicality of the wave function itself onto the real world it’s only the amplitude of the wave function that has implications for the real world so the wave function describes oscillating probabilities and it’s the the amplitude squared that tells us the probability per unit distance per unit time for something to be true in the shringer wave equation describing a matter wave involving either no forces or some forces but the wave nature of matter ultimately imposes a limit of absolute knowledge on our ability to understand the world around us what we learn from exploring the wave part of the wave function of the matter waves is that there’s a limit to our knowledge if we know the position of this wave very well we lose control over its momentum if we get control over its momentum at a high degree we lose our confidence in information about the position of the particle any longer these pair of variables are related to each other in their uncertainty by the Heisenberg uncertainty principle and fundamentally this imposes a limit of absolute knowledge on what we can know about a system of particles at any given moment in time by making measurements these are the foundations of quantum mechanics that we will build on going forward and we will spend the rest of this course essentially applying quantum mechanics and special relativity to problems involving the very small things in the universe like atoms and individual subatomic particles to make predictions about the natural world and understand phenomena like atoms and the behavior of particles trapped in systems like you would find for instance in semiconductors these are all basic applications that are at our fingertips now that we have a foundational equation that we can solve in order to understand the outcomes of these particular [Music] situations in this lecture we will learn the following things we will learn about the postulates of quantum mechanics the inviable tenets that are the foundations of this branch of physics we’ll also learn about some guidelines that you can employ for wave functions so that you can learn to solve the Schrodinger wave equation we’ll learn about classical analoges of quantum systems that we might want to model building on what we know already about a classical system but employing that in the Schrodinger wave equation and finally we’ll learn about a specific archetypal model a Quantum model of a bound particle known as the particle in a box or infinite Square well model and we will solve it using the Schrodinger wave equation let me remind you first about the one-dimensional Schrodinger wave equation which I will represent using a Shand going forward swe much easier to carry that around than shringer wave equation the schinger wave equation has a Time dependent statement on the left and on the right it has a spatially dependent statement about the wave function and finally it has a portion here that describes the action of an external force on the particle or system represented by the wave function the above is the one-dimensional schinger wave equation and generally speaking it allows for solutions that vary in space and time and it also allows for forces represented by the underlying potential that gives rise to the force that varies in Space in Time this is very complex so to utilize this equation we will need to do the following first we will represent physical situations with a model and what that usually boils down to because the time piece on the left and the space piece in the middle are essentially fixed by the form of the equation is varying the form of the potential V this describes how the system constrains particles described by the wave function now this effort may involve simplifying assumptions in the aid of creating a simple model of the force or forces that can act on the particle and these choices these simplifying assumptions have consequences that I’ll talk about later we will Define the base rules of quantum mechanics what are the inviolable tenants of problem solving in quantum mechanics that if untrue mean the fundamental dissolution of quantum mechanics we’ll also Define some guidelines for how to write down wave functions that will work to solve the shring or wave equation for instance in a specific situation now these guidelines may be violable depending on how you approximate physical situations but don’t represent of fundamental failure of quantum mechanics if violated in other words poor assumptions on the part of the problem solver the physicist are not to be held against the fundamental framework of quantum mechanics so what are the inviable tenets of quantum mechanics well these are known as the postulates of quantum mechanics and I’m going to warn you at the beginning that I am glossing over some of the Elegance of these postulates in favor of a bit more wordiness because we don’t have the mathematical foundations quite yet in order to take advantage of the more elegant and direct way of stating these postulates so what are the postulates of quantum mechanics well the first one is that at each specific time the state of a system that is for instance a particle or collection of particles can be entirely represented by a space of functions that are related to the wave function s now while sigh depends on a finite number of things like spatial position along the horizontal space axis and time the space of functions that can be related to the wave function and can fully represent the possible state of a system can be infinite in dimension now for our purposes we will concentrate just on the wave function rather than on this larger notion of a space of functions that can describe a system a more advanced course will concentrate rather on that space of functions which has all kinds of properties and rules associated with it it’s called a Hilbert space and it’s named after mathematical physicist David Hilbert the second postulate is that every observable quantity of a system for instance a measurement of momentum or energy will be represented mathematically by the action of an operator on the state of the system now I’ll elaborate more on on this a little bit later but think back to how I waved my hands and derived the shring or wave equation for example the total energy is measured in that equation by a Time derivative acting on the wave function and as you’ll see other actions of other derivatives effectively represent operators that measure quantities of the system these would be the outcomes of doing experiments and finally the only possible results of a measurement of an observable are related to characteristic numbers known as igen values of those

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