Phys621 — Graduate Quantum Mechanics I

Purpose of this course

Ideally this class will achieve several things:

• approach Quantum Mechanics somewhat more rigorously than in your undergraduate classes, using more sophisticated mathematical tools
• build a stronger conceptual understanding — the first time around you might have been preoccupied getting comfortable with the mathematics, such that you missed some of the 'big picture'
• begin to extend into more advanced areas so that you are ready to apply Quantum Mechanics to contemporary research projects

I will try to make the course as self-contained as possible, avoiding relying too much on any prior knowledge of Quantum Mechanics you might, or might not, have.

Class Schedule

Tuesdays and Thursdays, 11am — 12.20pm in Small Hall 235.

Office Hours

Thursdays 1.30pm — 3.00pm in Small Hall 328C.

Lecture Notes

I’ll post lecture notes on this website as we go along, which will sometimes contain a bit more information than we cover during lecture time. Hopefully this means you can do less writing and be a bit more focussed during lecture time. Please let me know of any errors you spot in the notes.

Problem Sets

There will be regular problem sets on the material we cover in lectures. These are a very important part of the course, almost certainly more important than listening to your lecturer waffle on. Sitting through lectures may make you feel like you have learned something, but you don’t really know until you try to use the techniques you think you have learned.

I will try my best to make the problems pedagogic so that you learn something by doing them. You should attempt the problems first on your own, but if you find you can’t solve a problem, you should seek help, either from me, or by collaborating with your classmates. But it is important that what you submit at the end represents your understanding of the problem — simply copying someone else's solution without understanding it is cheating and will not be tolerated.

You should try to present your solutions 'professionally' — they should feature text explaining what each major step in your solution is trying to do, and labelling any prior results you are making use of. I’ll provide my own solutions each week so you can get a better idea of what I mean. You don’t need to use LaTeX or other typesetting (unless you want to), but I do need to be able to read what you submit, so please think about legibility.

I want to emphasize that problems sets are not meant primarily as an assessment exercise, and for that reason, relatively little grade-credit is assigned to them. The payoff for putting in the effort comes in the form of learning, and the grade payoff comes when you get high grades in the midterm and the final exam, because you are so well-prepared, and then ace the QM problems on the qualifier.

The problem sets and deadlines are posted below.

Exams

Midterm exam in usual class time on October 24th.

Books

There is no single recommended QM book for this class. If you look you will find that about 99% of physics professors seem to write a QM book at some point during their careers, so there is a huge range to choose from. I’ll just list some of my favorites, with a few comments for each:

• Sakurai Modern Quantum Mechanics. Newer editions have been updated by Jim Napolitano. A nice and relatively short book at the graduate level. Clear notation, nice problems, but not a huge consideration of 'applications', and relatively little on wave equations.
• Gottfried & Yan Quantum Mechanics: Fundamentals. Gets through the basic material quite quickly, more formal in some places, but quite a few nice applications. Assumes more math confidence than Sakurai. More typos that you might like, but most of them are listed on a website.
• Landau & Lifshitz Quantum Mechanics (Non-relativistic theory). Hardcore, assumes you are a master of mathematics. But very thorough, and well written / translated from Russian. Notation may take some getting used to.
• Gasiorowicz Quantum Physics. Short book nominally at the undergraduate level, but a very nice read. A good start point if you've had little QM previously.
• Feynman The Feynman Lectures on Physics Vol III. Designed by Feynman to be taught to undergraduate freshmen! Not formal at all, assumes very little mathematics. Beautifully written, and fun to read, but not really a textbook in the usual sense of the word.

There are loads of other good books that I just don’t happen to know well, so find one that works for you. Most textbooks will say in the authors' foreword if they are aimed at the graduate level, and generally they all contain the same core topics. I would strongly recommend you do have at least one textbook on hand, and that you read the relevant sections in parallel as we go through the course.

Math

Familiarity with simple linear algebra is a huge help. You will probably be seeing plenty of this in PHYS603, so I won’t devote a lot of time to it. Some basic techniques for solving ordinary differential equations will also come in handy. Some of the more advanced methods we’ll consider are expressed in terms of complex variables, so some knowledge of properties of analytic functions and contour integration is helpful. Again, you'll likely see a lot of this in PHYS603 if you aren’t already familiar.

Topics

• Quantum behavior, interference and probability amplitudes — a pedagogic illustration
• Inferring the "rules" of quantum mechanical amplitudes using a simple experiment
• The mathematical structure of quantum mechanics
• States and probabilities
• Position and momentum
• Time evolution of quantum systems
• The Schroedinger wave equation in one-dimension
• Periodic potentials in one-dimension
• Rotations and angular momentum
• Three dimensional systems
• Coupling to electromagnetism

Grading

Problem Sets: 25%, Midterm Exam: 25%, Final Exam: 50%.

Next Semester

There is a second semester of this class in which we'll develop the theory of quantum mechanics further including topics such as approximation methods, coupling of angular momentum, scattering in three dimensions, many particle systems, path integrals, and attempts at constructing relativistic wave equations.

Lecture Notes

Problem Sets