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Physics 200: Relativity and Quanta


Syllabus

Grading scheme and policies

Information given here is approximate and subject to change until the start of term.

Pre-lecture preparation quizzes
3%  
Required pre-lecture study material and the related quiz will be posted on Connect by Thursday and due before the lecture on Monday. The quizzes are open book, but must be completed individually. Quizzes might contain open-ended questions with no 'right' answer, (such as 'What was the most confusing thing about the reading material?'). For these questions, you will need to write something sensible to be awarded marks. A grade of 80% on the pre-lecture reading quizzes will be enough to obtain 100% credit for this part of the course.
Clicker participation
3%  
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
4%  
During the tutorial, you will work in 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 for the worksheet as long as you make a serious attempt at answering the questions. Clickers might be used in the tutorial as an instructional aid, but grades will be assigned based on the handed-in worksheets only. Your lowest tutorial mark will be dropped when computing the final grade.
Problem Sets
20%  
Problem Sets will be assigned weekly, and you will have a week to complete each. The Problem Sets will be posted on Connect and will contain a mixture of multiple choice questions to be answered on-line and longer problems to be handed in 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.
Midterm 1
15%  
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
15%  
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

Quantum ereaser experiments, nuclear and particle 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:

Topic Specific Goals:

After taking the course, students should be able to:

RELATIVITY

Newton's Laws and Relativity

Puzzles from Electromagnetism

Einstein's Resolution

Relativistic Invariants

Relativistic Energy and Momentum

QUANTUM MECHANICS

The electromagnetic description of light

Problems with classical physics

Light as a Particle

Properties of Quanta of Light ("Photons")

The quantum description of electrons

Momentum eigenstates, wavepackets, and uncertainty

The Schrödinger Equation

Bound states and atomic spectra

Tunneling