Peter Woit (woit@math.columbia.edu)

Mathematics 421

Tuesday and Thursday 2:40-3:55pm

407 Mathematics

First four classes currently planned to be on Zoom, links available at the Courseworks site for this class.

For the first two weeks I won't schedule office hours, but will be available in Mathematics 421 most times, feel free to stop by, or email and set up a meeting time. Also email if you would like to schedule a meeting on Zoom.

Teaching Assistant: Zoe Himwich (zmh2110@columbia.edu)

This course will be an introduction to the subject of quantum mechanics, from a perspective emphasizing the role of Lie groups and their representations. Most of the standard material and examples from a conventional first physics course on the subject will be covered, but with much greater attention to the mathematical ideas behind the standard formalism and usual calculational techniques.

No specific background in physics will be assumed, although an elementary physics course of some kind would be helpful. The mathematical prerequisites are multi-variable calculus (as in Calculus IV), and Linear Algebra. This course is open to both undergraduate and graduate students. It can be taken independently and in addition to any of the Physics department courses on quantum mechanics.

The lecture notes from previous versions of this course have been
turned into a book, see here.
During this course I expect to be revising some of the material in
the book, and maybe adding some new chapters. The most
recent version will always be available here.

During the semester I expect to cover roughly the material in the first 23 chapters of the book. Before each class, please try and read the chapter in the syllabus announced for that class and come prepared with questions about whatever you don't understand. I hope to devote much of the time in each class to going over material students are finding confusing, rather than repeating everything that is in the notes.

Last time I taught the class, it was online-only because of COVID
and videos are available
on Youtube.

First problem set: due Tuesday, Feb. 1 .

Problems 1-4 in appendix B.1 and Problems 1-4 in appendix B.2 of the book

Second problem set: due Thursday, Feb. 10.

Problems 1-3 in appendix B.3 of the book

Third problem set: due Tuesday, Feb. 15.

Problem 4 in appendix B.3

Fourth problem set: due Tuesday, Feb. 22.

Problems 1-4 in appendix B.4 and problem 1 in appendix B.5 of the book

Fifth problem set: due Tuesday, March 8.

Problems 1-2 and 5 in appendix B.6 of the book.

Sixth problem set: due Tuesday, March 22.

Problem 3 in appendix B.6 of the book. Problem 4 in appendix B.6 optional.

Seventh problem set: due Tuesday, April 5.

Problems 1-4 in appendix B.7 of the book.

Eighth problem set: due Tuesday, April 12.

Problem 1 in appendix B.8 of the book.

Ninth problem set: due Tuesday, April 19.

Problems 1-3 in appendix B.9 of the book.

Tenth problem set: due Thursday, April 28.

Problems 1-4 in appendix B.10 of the book.

Chapter numbers correspond to the course textbook,

Tuesday, January 18: Introduction and overview (Chapter 1)

Thursday, January 20: The group U(1) and charge (Chapter 2)

Tuesday, January 25: Two-state systems and spin 1/2 (Chapter 3)

Thursday, January 27: Linear algebra review, orthogonal and unitary groups (Chapter 4)

Tuesday, February 1: Lie algebras and Lie algebra representations (Chapter 5)

Thursday, February 3: Rotations and spin in 3 and 4 dimensions (Chapter 6)

Tuesday, February 8: The spin 1/2 particle in a magnetic field (Chapter 7)

Thursday, February 10: Representations of SU(2) and SO(3) (Chapter 8)

Tuesday, February 15 : Tensor products (Chapter 9)

Thursday, February 17: Review

Tuesday,
February 22: Midterm exam (material through Chapter 9 of the
notes)

Thursday, February 24: Momentum and the free particle (Chapter 10)

Tuesday, March 1: Fourier analysis and the free particle (Chapter 11)

Thursday, March 3: Position and the free particle (Chapter 12)

Tuesday,
March 8: The propagator for a free particle (Chapter 12)

Thursday,
March 10: The Heisenberg group and the Schrödinger representation
(Chapter 13)

Tuesday, March 22: The Poisson bracket and symplectic geometry
(Chapter 14)

Thursday, March 24: Quadratic polynomials and the symplectic group
(Chapter 16)

Tuesday, March 29: More on quadratic polynomials and the
symplectic group (Chapter 16)

Thursday,
March 31: Hamiltonian vector fields and the moment map (Chapter
15)

Tuesday, April 5: Quantization (Chapter 17)

Thursday, April 7: The Euclidean group (Chapter 18)

Tuesday, April 12: Quantum free particles and representations of
the Euclidean group (Chapter 19)

Thursday, April 14: Central potentials and the hydrogen atom
(Chapter 21)

Tuesday, April 19: More on the hydrogen atom

Thursday, April 21: The harmonic oscillator (Chapter 22)

Tuesday, April 26: More on the harmonic oscillator

Thursday, April 28: Review

Final exam is scheduled for Thursday, May 12, 1:10-4pm

Principles of Quantum Mechanics, by Ramamurti Shankar. Springer, 1994.

which does a good job of carefully working out the details of many calculations. Two good undergraduate-level texts are

A Modern Approach to Quantum Mechanics, John S. Townsend, University Science Books, 2000.

Introduction to Quantum Mechanics, David J. Griffiths, Prentice-Hall, 1995.

Several suggestions for standard physics textbooks that provide good references for some of the topics we(The following is from when I last taught the course, will be updated soon for the spring 2022 version). will be considering are:

Quantum Mechanics, Volume 1, by Cohen-Tannoudji, Diu and Laloe. Wiley, 1978

The Feynman Lectures on Physics, Volume III, by Richard Feynman. Addison-Wesley 1965.

Lectures on Quantum Mechanics, Gordon Baym.

Quantum Mechanics, Volumes 1 and 2, Albert Messiah.

Quantum Mechanics, Volume 1, Kurt Gottfried.

Introduction to Quantum Mechanics, David J. Griffiths.

Quantum Mechanics and the Particles of Nature: an Outline for Mathematicians, Sudbery. Cambridge 1986 (unfortunately out of print)

Some other books at various levels that students might find helpful:

More mathematical:

An Introduction to Quantum Theory, by Keith Hannabuss. Oxford, 1997.

Quantum Mechanics for Mathematicians, by Leon Takhtajan. AMS, 2008.

Lectures on Quantum Mechanics for Mathematics Students, by L.D. Fadeev and O.A. Yakubovskii. AMS, 2009.

Linearity, Symmetry and Prediction in the Hydrogen Atom, Stephanie Singer, Springer, 2005. (On Springerlink at this URL)

Some more from the physics side, available via Springerlink:

Quantum Mechanics, Franz Schwabl.

Lectures on Quantum Mechanics, Jean-Louis Basdevant.

Quantum Mechanics, Daniel Bes.

A classic:

The Theory of Groups and Quantum Mechanics, Hermann Weyl.

Also emphasizing groups and representations, but covering mostly different material:

Group theory and physics, Shlomo Sternberg.

More advanced, from the point of view of analysis:

Mathematical Methods in Quantum Mechanics, Gerald Teschl

Recommended sources on Lie groups, Lie algebras and representation theoy:

Groups and Symmetries: From Finite Groups to Lie Groups, Yvette Kossmann-Schwarzbach

An Elementary Introduction to Groups and Representations, Brian C. Hall

Lie groups, Lie algebras and representations, Brian C. Hall

Representation Theory, Constantin Teleman

For more about Fourier analysis, see notes from my Spring 2020 Fourier analysis class.

Online Resources

Lecture notes for a course on Quantum Computation, John Preskill (especially Chapters 1-3)

Introduction to Quantum Mechanics, Fall 2012: Math W4391

Introduction to Quantum Mechanics, Spring 2013: Math W4392

Introduction to Quantum Mechanics, Fall 2014: Math W4391

Introduction to Quantum Mechanics, Spring 2015: Math W4392

Introduction to Quantum Mechanics, Fall 2020: Math GU4391

Introduction to Quantum Mechanics, Spring 2021: Math GU4392