Math 480C, Spring 2008

Office Hours: Tentatively, MW 4-5 and F 1:30-2:20. This could change. You can also make appointments for other times.

Text: Linearity, Symmetry, and Prediction in the Hydrogen Atom by Stephanie Frank Singer.

What the course is about: The aim will be to cover the first eight chapters (at least) of the text. After some general introductory material, they give an introduction to the theory of the rotation group (together with some closely related groups) and its representations, with the goal of understanding (some of) the physics of the hydrogen atom.

I do not want to follow the usual lecture-homework-exam format. I want the course to be more interactive, with the students doing a lot of talking to me and to each other. There will be homework problems, some of which will probably be handed in as "group projects," but much of the assigned work will consist simply of reading the book in detail (plus perhaps some supplementary material now and then). A lot of class time will be devoted to discussions of (rather than lectures on) the body of the book and the exercises.

There will be a term paper. Everybody gets to choose their own topic, subject to approval; it can be any one of a lot of topics in math and/or physics related to the material of the course. Click here for more details.

I am not planning to give any exams.

Corrections to the text: I am going to compile a list of corrections (and other comments) to Singer's book as we go along. If you see something in the book that seems to be a mathematical or typographical error, let me know. To access the list, click here: Corrections Page

Assignments: (Homework to be handed in will be indicated explicitly.)

• For Wed 4/2: Read Sections 1.1--1.5, do Exercises 1.5, 1.6.
• For Fri 4/4: Read Section 1.7, do Exercise 1.20.
• For Mon 4/7: Read Sections 2.1-2.5. This should be review, but I am happy to answer questions about it in class or otherwise, so don't be afraid to ask. Look at Exercises 2.3, 2.4, 2.6, 2.7, 2.13, 2.19, 2.26. I hope your reaction to all of these will be "Oh yeah, this is obvious" or "Oh yeah, this is very familiar." But maybe not, so ask questions if necessary! Also, I'm assigning a couple of hand-in problems for next Wednesday:
• For Wed 4/9: Hand in Exercise 1.25 and Exercise 2.23 (as corrected on the second errata page). Notes on Exercise 1.25: (i) You can replace 3 by n; this works the same way in any dimension. (ii) Hint #1: The dot product v.w of two vectors is the matrix product (v^T)w if you think of v,w as column vectors and v^T is the corresponding row vector. Hint #2: You can express dot products in terms of norms by the formula 4v.w = |v+w|^2 - |v-w|^2.
• For Fri 4/11: Read Sections 3.1--3.3 (together with the comments on them that I have put on the Corrections Page). Think about Exercises 3.6 and 3.13; we'll talk about them in class on Friday. In 3.13, take W to be a subset of V, not necessarily a subspace.
• For Mon 4/14 and Wed 4/16: Read Sections 3.4 and 3.5. Do the problems on the sheet I handed out; a copy of it is here.
• For Fri 4/18: Prepare a prospectus for your term paper.
• For Mon 4/21: Read Sections 4.1-4.3. Look at Exercises 4.1, 4.3, 4.7, 4.11. Also note the recent additions to the correction sheet.
• For Wed 4/23: Think about Exercises 4.7, 4.11, 4.12 for discussion in class. (If you can figure out how to do 4.7, congratulations. 4.11 and 4.12 are easier.)
• For Fri 4/25: Do the hand-in problems on the handout sheet.
• For Mon 4/28: Read Sections 4.4-4.6. Do Exercises 4.21, 4.22, 4.24.
• For Wed 4/30: Read Sections 4.7, 5.1-2. Look at Exercise 5.17.
• For Fri 5/2: Do the hand-in problems on the handout.
• For Mon 5/5: Browse through the rest of chapter 5 (which we won't cover in detail). Look at Exercise 5.17 again. (The result referred to in the last part is Prop. 5.7.)
• For Wed 5/7: Read Sections 6.1-6.3 and my guide to them.
• For Fri 5/9: Read Sections 6.5-6.6. Hand in the problems on the last page of the guide to Chapter 6.
• For Mon 5/12: Read Sections 7.1-7.2 together with the supplement that I handed out on Friday.

My notes on the mathematical structure of quantum mechanics (corresponding approximately to the lecture on 4/16) can be downloaded here.