Physics 200: Relativity and Quanta

2010 Course webpage

Grading scheme and policies

Pre-lecture reading quizzes	3%	The quiz must be started no less than an hour before the lecture to which it corresponds. Quizzes will be available on WebCT Vista at least a week before they are to be completed. Reading assignments for each lecture will be posted in advance on the Course Content page. The quizzes are open book, but must be completed individually. Your two lowest quiz marks will be dropped when computing the final grade.
Clicker participation	4%	Clicker questions will be asked in lectures. To encourage discussion, your grade will depend on the number of questions you participate in, not on whether or not your answers were correct. You will need to answer at least 80% of all questions to receive full marks.
Tutorial participation	5%	During the tutorial, you will work is small groups but will be required to complete your own tutorial worksheet. You will be asked to hand in your tutorial worksheet at the end of the tutorial. You will be given full marks as long as you make a serious attempt at answering the questions. Your lowest tutorial mark will be dropped when computing the final grade.
Problem Sets	20%	Problem Sets will be assigned roughly weekly, and you will have a week to complete each. They will be handed out in class and posted on the course website. They will be typically due in class; if you cannot make it to the class the Problem Set is due on, you need to hand it in ahead of time. Late (even by an hour) Problem Sets will not be accepted for credit under any circumstances, since solutions will be posted immediately after the due date. If you do not hand in your Problem Set on time, you will get a zero on that Problem Set. In exceptional circumstances the zero you get on a late Problem Set will not count towards your grade. I will require advance notice of such circumstances or a proof of the emergency (doctors notes and police records are good proofs), and you must still finish the Problem Set. Excuses such as `I was really busy with other courses' will not be accepted. Your lowest Problem Set mark will be dropped when computing the final grade. Group discussion of Problem Sets is encouraged, but the solutions you hand in must be your own work. This means you should not be looking at anybody else's notes or assignment while writing up your solutions and you should not share your completed Problem Set with anybody else. Problem Sets will be graded using the attached grading rubric.
Midterm 1	14%	The two midterms will be written in-class. Check the Course Content page for scheduling. There will be no make-up midterms and tests. If you miss a midterm for a valid reason (proven illness, a family emergency), the weight of the missed midterm will be reassigned to the other midterm and the final exam. If a scheduled test falls on one of your religious holidays, please let me know as soon as possible so that I can make alternative arrangements. A notice of at least two weeks is required.
Midterm 2	14%
Final exam	40%	Since many of the learning goals for this course are conceptual, a significant portion of the exam (and midterms) will be made up of (mostly multiple choice) questions requiring little or no calculation. The reminder of the exam will be problems.
Total	100%

UBC takes academic misconduct (this includes copying of homework, cheating on exams and plagiarism) very seriously, and the penalties are stiff. Please check pages 48-49 and 54-55 of the calendar for official university regulations.

Course outline

The outline below should help give you the big picture of what we'll discuss in the course. You may want to refer back to this during the course to see how the current topic fits into the big picture, and also see where we're going. Some of the sections will probably make more sense after we have begun discussing a topic.

PART I: SPECIAL RELATIVITY

Newtons laws and relativity

To begin, well review Newtons laws and point out that if these are true for one person, they are equally true for anyone else moving at a constant velocity relative to that person. So two people moving at some constant relative velocity will have exactly the same laws of mechanics. This means that if the two people set up identical (mechanical) experiments, they will get identical results. It is then impossible to come up with any mechanical experiment to measure absolute velocity, since such an experiment would have to give different results for people moving at different velocities. Thus, at least from the point of view of mechanics, only relative velocities have any practical meaning, and this is what is meant by the principle of relativity.

Puzzles from electromagnetism

Well then review some basic electricity and magnetism and recall how light arises as an electromagnetic wave. Since the equations of electromagnetism (Maxwells Equations) predict a specific value for the speed of light, it sounds like the principle of relativity must be violated for electricity and magnetism: according to the usual rules of adding velocities, if the speed of light is X kilometers per hour as measured by one person, it would be X + v kilometers per hour as measured by a person moving toward the light source with velocity v kilometers per hour. If this is correct, only one of these two people could possibly observe the value predicted by electromagnetism. We could then define a stationary observer as one for whom Maxwells Equations (and their prediction for the speed of light) hold. This would give a practical distinction between moving observers and stationary observers, and an absolute notion of velocity.

Einsteins resolution: special relativity

Einstein didn't like this conclusion, and various experiments, notably one performed by Michelson and Morley, could find no evidence that the speed of light was different for different observers. So Einstein proposed that the principle of relativity does hold for electricity and magnetism. He realized that this postulate is self-consistent, but that it requires some profound revision of our understanding of space and time. Einsteins theory, now known as Special Relativity, implies that two observers moving at some large relative velocity will not agree on the time intervals or distances between events, or even whether two events occur at the same time or at different times. To understand this, well have to think carefully about how distances and times are measured and how the measurements as performed by one observer are related to those performed by another observer moving at some constant relative velocity. With the basic assumption that all observers should agree on the speed of light (in a vacuum), well see that there are unique, mathematically precise, rules for relating measurements made by observers moving at different velocities. These agree with our ordinary intuition in the case where all velocities are much less than the speed of light, but give rise to the startling consequences we have mentioned in cases where velocities become large. Despite these counterintuitive results, the new rules provide a consistent framework for physics involving arbitrary velocities, and are now supported by compelling experimental evidence that we will discuss.

Relativistic invariants

In order to talk about physics in the new framework, we will want to understand which quantities two observers moving with some relative velocity will agree upon, since these are the quantities that they can sensibly compare with each other. We will see that there are invariant (i.e. the same for all observers) notions of distance, time and simultaneity (called proper distance, proper time, and spacelike separation respectively) that generalize our usual notions. Well see that many of the new concepts can be understood most clearly in a pictorial way using spacetime diagrams.

Relativistic energy and momentum

After understanding the new framework for measuring and comparing lengths and times well see that the usual definitions of momentum and energy will have to be modified in order that the conservation of energy and momentum still hold. Well see that the correct definition of energy includes a term that is non-zero even for zero momentum (and zero potential), namely the mass times the speed of light squared. It is only the combination of this mass energy and the energy associated with momentum that is conserved in relativistic processes, so it is possible to convert mass energy to kinetic energy and vice-versa. With this observation, we can understand why the mass of a hydrogen atom is less than the mass of an electron plus the mass of a proton, and why nuclear reactions can be used to produce enormous amounts of energy. Finally, well see that with the new definitions, energy and momentum can be non-zero and finite even in a limit where the mass is taken to zero, assuming that the velocity is taken to be that of light. Furthermore, the relationship between energy and momentum for massless particles is exactly the same as for classical electromagnetic waves. We will soon see that the similarities between massless particles traveling at the speed of light and electromagnetic waves are not a coincidence.

PART II: QUANTUM MECHANICS

Light as a particle

To introduce quantum mechanics, well begin by pointing out a few simple phenomena that classical mechanics and classical electromagnetism cannot seem to explain. One of these, the photoelectric effect (in which electromagnetic radiation liberates electrons from a metal), suggests strongly that light comes in discrete bundles or quanta of energy, with the energy in each quantum proportional to the frequency. We can think of these quanta as particles of light, called photons, which together make up the electromagnetic wave.

Properties of quanta

If the photon description of light is correct, we should be able to explain wavelike phenomena via the behavior of individual photons. We will quickly realize that this is only possible with some drastic departures from the rules of classical physics. To begin, well discuss the photon interpretation of familiar experiments involving polarizers. For photons of light polarized in the same direction as a polarizer or perpendicular to the polarizer, it must be that all the photons pass through or none of the photons pass through respectively. However, in order to explain the partial attenuation observed for any other polarization, we will be forced to conclude that for a stream of these identically polarized photons, some pass through the polarizer and some do not. Even with complete knowledge about the initial polarization, we cannot predict the fate of any individual photon, only the probability that it will pass through the polarizer. This indeterminacy is a central difference between quantum mechanics and classical mechanics: whereas in classical mechanics, we could hope to predict the precise future evolution of a system, in quantum mechanics, we can only predict the probabilities for the various possible outcomes of an experiment. In the classical picture, light waves with general polarizations are superpositions of waves with orthogonal linear polarizations. To quantitatively explain the polarizer experiments based on photons, we will argue that the individual photons with general polarizations should still be viewed mathematically as superpositions of the two special photon states, but one in which the overall amplitude has no physical meaning. Thus, well arrive at one of the most basic rules for quantum systems: for each possible result of a given measurement, there are special states, called eigenstates for which that result will definitely be obtained. More general states (for which the result is not predetermined) are superpositions of these eigenstates, and the amount of each eigenstate in the superposition determines the probability of obtaining the corresponding outcome.

Wave properties of particles

With the understanding that classical electromagnetic waves are comprised of photon particles, one might wonder whether other kinds of particles give rise to wavelike phenomena. While we don't see any classical electron waves (this has to do with the Pauli exclusion principle) it turns out that a beam of electrons at some fixed momentum does exhibit diffraction phenomena, with a wavelength inversely proportional to momentum, just as for photons. By discussing a very simple diffraction experiment, known as the double-slit experiment, well argue that states of individual electrons with a given momentum do not have well defined positions, and propose that these and more general states of the electrons are superpositions of eigenstates where the electrons do have definite positions, motivated by our discussion of polarization experiments. The information about the amount of each position eigenstate in a given superposition is known as the wavefunction, and this information determines the probabilities of the possible outcomes when the position is measured. The wavefunction gives a complete description of the state of a particle at a given time and replaces classical description in terms of instantaneous position and velocity. The description of general electron states as superpositions of states with definite positions provides another example of our general rules for quantum mechanics. For any given question that we might ask about a particle (in this case what is the position? or what is the momentum?), there are some states for which the answer is predetermined, while general states are superpositions of these states. An important point is that an eigenstate for one physical quantity (e.g. an electron with some definite position) is usually not an eigenstate for another physical quantity. For example, there are no states that are eigenstates of both position and momentum, and well see that the more certain we are about the position of a given particle, the more uncertainty there is in the momentum. This is known as the Heisenberg Uncertainty Principle.

The Schrödinger Equation

Since the classical description of a particle in terms of position and velocity have been replaced by the idea of a quantum state described by a wavefunction, well need to understand what replaces Newtons Second Law and determines how the wavefunction evolves with time. The diffraction phenomena observed for electrons suggests that electron states with definite momenta should behave like propagating waves with wavelength inversely proportional to the momentum and frequency proportional to the kinetic energy. By postulating that the wavefunctions for momentum eigenstates behave in this way, we will arrive at an evolution equation for wavefunctions known as the Schrödinger equation.[1] This is the general equation for time evolution of quantum states, and the remainder of the course will be spent exploring its consequences.

Bound states and atomic spectra

We will first study the Schrödinger equation for particles which are classically trapped in a finite region of space due to external forces. An example is an electron in an atom whose energy is less than the amount required to overcome the Coulomb attraction of the nucleus. We will see that in these cases, the Schrödinger equation implies that the particle can exist only at certain specific energies. This explains the discrete nature of atomic spectra: electrons can absorb or emit energy only in the precise amounts that allow them to jump between their allowed energies.

Tunneling (time allowing)

One of the surprising consequences of the Schrödinger equation is that particles have some probability of being found in places where the classical potential energy is greater than the total energy of the particle. A result of this is that particles can pass through barriers created by external forces which would classically block them completely. This phenomenon can be used to understand certain types of radioactive decay and is central to a number of important technological applications, such as scanning-tunneling electron microscopes that are able to see individual atoms.

POSSIBLE SPECIAL LECTURES

Nuclear and particle physics, statistical mechanics and condensed matter physics, general relativity and cosmology

[1] A more complete derivation for the Schrödinger Equation is possible, but requires a more in depth treatment of quantum mechanics, together with some relatively advanced results in classical mechanics.

Learning Goals

The following is a list of learning goals for the course. These are things you should know or be able to do once we have covered each topic.

Broad Goals:

After taking this course, students should be able to:

state the principle of relativity, and be able to describe some of the basic implications of this that go against our usual intuition (and explain how experimental evidence supports these)
analyze simple dynamical processes using relativistic dynamics. This will include:
- knowing how to relate physical quantities measured by observers moving at some relative velocity and know which quantities observers at relative velocities will agree upon.
- knowing the limits of applicability of elementary formulae from mechanics and know the more general formulae that replace these in situations where velocities are not a negligible fraction of the speed of light
describe and predict basic behavior of light and electrons in terms of both classical mechanics and quantum mechanics descriptions, and be able to specify differences
argue how the assertions of quantum mechanics are inferred from the experimental evidence
calculate the characteristics of quantum states and probabilities for outcomes of measurements for a few very mathematically simple quantum systems, including ones that the student has not seen before. This requires understanding the basic mathematical framework well enough to apply it to such novel systems.
give qualitative predictions and explanations of the behavior of simple quantum systems, such as the distribution of electrons in atoms and the spectrum of light emitted and absorbed by atoms
argue that physics goes beyond a collection of empirical laws, and involves a deeper conceptual framework that is inferred from experiment but is not at all obvious from our everyday experience
better understand popular science articles on current research in physics and answer questions about modern physics from curious friends and relatives
see value in achieving a deeper understanding of quantum mechanics and learning more about modern physics

Topic Specific Goals:

After taking the course, students should be able to:

RELATIVITY

Newton's Laws and Relativity

explain what is meant by the principle of relativity
explain what is meant by a `frame of reference' and an `inertial frame'
give examples of how relativity manifests itself in ordinary situations
show that Newton's Laws obey the principle of relativity
explain why the equivalence of physical laws in different frames implies that it is impossible to set up an experiment to measure an absolute velocity

Puzzles from Electromagnetism

explain why the principle of relativity applied to electromagnetism implies that the speed of light should be the same in all frames of reference
give simple examples from electricity and magnetism to show that either the principle of relativity or some basic notions of distance, time, and velocity must be abandoned

Einstein's Resolution

describe experimental evidence that suggests the velocity of light is always independent of the of the motion of the source or of the observer
state Einstein's principle relativity
explain how a given observer can set up a coordinate system for making measurements of time and position
be able to describe what is meant by and event or a trajectory
be able to show how Einstein's postulates imply that observers at large relative velocities will not agree on distances, time intervals or whether two events are simultaneous
describe qualitatively the meaning of length contraction, time dilation, and the relativity of simultaneity
correctly calculate the lengths and times differences that an observer will measure, properly accounting for length contraction and/or time dilation.
use Lorentz transformation formulae to relate the measurements of observers moving at relative velocities
determine the trajectory of an object in one frame of reference given its trajectory in another frame of reference
analyze basic scenarios involving large velocities to calculate times and distances for various events, or physically relevant time/distance intervals. Know when basic length contraction and time dilation formulae are applicable and when Lorentz transformations are needed.
use the velocity transformation formula to calculate the observed velocity of an object in a new frame given the velocity in the old frame and the relative velocity of the two frames
calculate the Doppler shift of light frequencies for an observer moving relative to the source and discuss the importance of the Doppler effect in astrophysics
be able to calculate relativistic effects in cases when velocities are much smaller than the speed of light, using Taylor (binomial) approximations to the exact formulae

Relativistic Invariants

describe the meaning of spacelike separation, timelike separation, proper length, and proper time
be able to determine whether two given events are spacelike, timelike, or lightlike separated; equivalently be able to determine whether there is a frame where two events are simultaneous or a frame where two events are at the same location
know how to calculate the proper length/time between two events and the time elapsed on the clock of an observer on some general trajectory
represent graphically simple scenarios on a spacetime diagram
use spacetime diagrams to analyze simple processes involving relativistic velocities and resolve apparent paradoxes (e.g. ladder passing through barn)

Relativistic Energy and Momentum

argue why classical formulae for momentum and energy must be modified
state the relativistic formulae for energy and momentum
be able to determine the energy and momentum in one frame of reference given these quantities in another frame of reference
explain the precise meaning of conservation of energy and conservation of momentum
analyze high-energy particle decay processes and scattering processes using energy and momentum conservation
provide a definition for mass in terms of energy, and apply this to make predictions about the masses of stable and unstable bound states relative to the masses of their constituents
determine the mass of an object given its energy and momentum
explain why the conservation of mass can be violated in relativistic dynamics
give evidence for and explain basic implications of the equivalence between energy and mass
state the relation between energy and momentum for massless particles and use this in analyzing dynamical processes involving light or other massless particles

QUANTUM MECHANICS

The electromagnetic description of light

describe the electric and magnetic fields inside a beam of light; explain what is meant by the wavelength, period and frequency in this description
state the relation between frequency and wavelength for light and explain why this must be true
explain how basic observable properties of light (colour, brightness, polarization) are related to details of the mathematical description as an electromagnetic wave (e.g. amplitude, wavelength)
describe how the energy density and momentum density in a light beam are related to the wavelength and the amplitude
explain precisely what is meant by the intensity of a light beam and how this is related to the other quantities associated with the classical electromagnetic description
describe the basic mechanism for producing electromagnetic radiation

Problems with classical physics

Explain why the ordinary mechanics and electromagnetism fail to explain the stability of atoms and the spectrum of light from a heated atomic gas

Light as a Particle

Describe the photon model of light and explain how the wavelength/frequency and amplitude/intensity of a light beam are related to the underlying properties of the photons
qualitatively describe what is measured in the photoelectric effect experiment and explain how this implies a quantum picture of light, including explaining what results the classical interpretation of light would predict for this experiment
explain why there is a maximum wavelength above which light cannot eject electrons from a metal regardless of intensity
explain the relation between the maximum kinetic energy of the ejected electrons and the frequency of the incident light, as predicted by the photon model
quantitatively analyze photoelectric data to deduce the relationship between energy of photons and frequency of light

Properties of Quanta of Light ("Photons")

describe what is meant by polarized light and predict the reduction in intensity that results from sending polarized light through a polarizer at some angle
argue why the photon explanation of the polarizer experiments demands a probabilistic ( i.e. non-deterministic) behavior of photons
predict the likelihood of various outcomes in simple experiments governed by probabilistic behavior
use the mathematical model of photon polarizations as unit vectors to predict probabilities for photons being transmitted through polarizers
define what is meant by an "eigenstate" for a given measurement in the context of photons passing through polarizers
explain how the probabilities for transmission are related to the details of how a photon state decomposes into a superposition of eigenstates for the polarizer
compare and contrast classical superposition and quantum superposition
describe how a photon's polarization state changes when it passes through a polarizer
explain how the simple model for calculating transmission probabilities has features that generalize to all quantum systems

The quantum description of electrons

describe the double slit experiment for light or electrons and explain why this provides evidence that quantum particles do not have definite positions and can exist in quantum superpositions
explain why the results of the double slit experiment imply that the initial electrons do not have well defined positions
explain why the double slit experiment suggests that the behavior of single particles is probabilistic and how the classical intensity pattern is related to the relative probability for hitting various points on the screen
Describe what is meant by probability density and evaluate whether or not a given function is a valid probability density for finding a particle
Explain what is meant by a position eigenstate and what is meant by a quantum superposition of position eigenstates.
explain how the process of describing general quantum superpositions of position eigenstates leads to the concept of a wavefunction
Use the wavefunction to determine the probability for finding a particle in a given region of space. Be able to determine the proper normalization of the wavefunction if it is not given.
State what is meant by the expected value (or expectation value) in a measurement of position. Be able to calculate the expected value given a particle's wavefunction
Explain what happens to the wavefunction after a measurement of the particle's position

Momentum eigenstates, wavepackets, and uncertainty

explain why the double slit experiment suggest that we can associate a wavelength to electrons with a particular momentum
state de Broglie's relationship between wavelength and momentum of an electron or other particle
write down the mathematical description of the wavefunction for a momentum eigenstate; explain how this can have a wavelength even though the probability density is constant
explain what is meant by a wavepacket and why the wavefunction for a real traveling electron should take this form
give a simple explanation for why particles with well defined momentum cannot have a definite position
explain the physical interpretation of the mathematical fact that wavepackets and other wavefunctions can be written as a sum of pure waves
use the amplitude function describing the superposition of pure waves contributing to a given wavefunction to predict the relative probability for possible outcomes in a measurement of momentum
explain qualitatively how the width of a wavepacket and its wavelength relate to the combination of pure waves (momentum eigenstates) that it is built from
state the Heisenberg Uncertainty Principle and explain how it is related to the previous mathematical fact
explain what is meant by uncertainty in position and uncertainty in momentum for a state
explain why we can't violate the Heisenberg Uncertainty Principle simply by making a measurement of momentum immediately after a measurement of position
use the Heisenberg Uncertainty Principle to predict how quickly a given wavepacket will spread out with time

The Schrödinger Equation

predict the velocity of a given wavepacket and how fast it will spread out by looking at the wavepacket at some initial time
explain the difference between phase and group velocities for a wavepacket
explain what is meant by a dispersion relation
know that the relation between frequency and wavelength for a specific kind of wave determines the relation between wavelength and group velocity
explain why the group velocity for electron wavepackets should be inversely proportional to wavelength and (qualitatively) how this leads to the relation between frequency and energy for electron wavefunctions
write down the time-dependent wavefunction for a momentum eigenstate and relate its frequency and wavelength to the momentum
explain why knowing the time dependence of momentum eigenstate wavefunctions allows us to determine the time-dependence for general wavefunctions (of a free particle)
explain why the Schrödinger equation determines the time-dependence of a wavefunction
explain why the Schrödinger equation can be interpreted as a relationship between wavelength and frequency for wavefunctions
write down the potential function for simple physical systems including electrons in wires or electrons near other charges

Bound states and atomic spectra

describe what is meant by an energy eigenstate, and state important properties that all energy eigenstates satisfy
know what is meant by the statement that energy eigenstates are `stationary'
Describe how we would check using the Schrödinger equation whether a certain initial wavefunction is an energy eigenstate with energy E, and how this leads to the time-independent Schrödinger equation
describe what is meant by a bound state (either in classical physics or in quantum mechanics)
explain the crucial difference between the allowed energies for bound states in quantum mechanics as compared to classical mechanics
describe how we would go about finding the possible energies for bound states for a quantum system described by some specific potential energy function
explain why properties of quantum bound states can explain why atoms are stable and why atomic spectra have discrete frequencies
be able to calculate the emission spectrum of an atom or molecule given its bound state energies (an vice versa)
be able to predict the wavelengths of light for which a gas of atoms will be opaque or transparent given its bound state energies
explain what is meant by zero point energy and why this energy is a consequence of the Heisenberg Uncertainty Principle
state the possible energies that might be obtained in a measurement performed on a simple superposition of energy eigenstates
explain qualitatively how the probability density for a superposition of energy eigenstates behaves differently from that of an energy eigenstate
predict probabilities for measurements of energy or position for simple superpositions of the energy eigenstates

Tunneling

describe the behavior of the wavefunction in regions where the particle is classically forbidden
describe the qualitative evolution of a wavefunction initially localized in one half of a double well
describe the evolution of a wavefunction for a particle that is traveling towards a potential energy barrier that it does not have enough energy to pass through classically
explain how tunneling can be used to understand radioactivity and to produce atomic scale "images" with scanning-tunneling electron microscopes