Introduction to Quantum Mechanics: Mathematics GU4391 (fall 2020)


Tuesday and Thursday 4:10-5:25pm

Class lectures will be online-only for now (NOT in Math 203 as listed by the registrar).  Zoom links are available on Courseworks.  As an experiment, I'll also have videos of the class available at this Youtube channel.

Teaching Assistant:  Davis Lazowski (lazowski@math.columbia.edu)
Davis will have office/help room hours Monday 5-6pm and Thursday 9-11am.

I'll have online office hours after each class starting around 5:30, there are separate Zoom links for the office hours on Courseworks.  I will be in the office most of the time Tuesday and Thursday. If you are in the area and able to come in for an in-person office hour, please contact me (or just try and stop by Math 421 Tuesday or Thursday).

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 conventional physics courses will be covered, but with much greater attention to the mathematical ideas behind the standard formalism and usual calculational techniques.   There will be a continuation of this course (Math GU4392) in the spring covering more advanced material.

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.

Lecture Notes/Book

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 fall 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.

Problem Sets and Exams

There will be problem sets due roughly every week, a midterm and a final exam. The final exam is scheduled for Tuesday, December 16, 4:10-7pm.  Use of notes is allowed during the exams.  Grading will be based on these according to: 50 % final exam, 25 % midterm exam, 25 % problem sets.

First problem set: due Tuesday, Sept. 22. 
Problems 1-4 in appendix B.1 and Problems 1-4 in appendix B.2 of the book

Second problem set: due Tuesday, Sept. 29.
Problems 1-3 in appendix B.3 of the book


Tentative Schedule of Lectures

Chapter numbers correspond to the course textbook, Quantum Theory, Groups and Representations.


Tuesday, September 8: Introduction and overview (Chapter 1)

Thursday, September 10: The group U(1) and charge (Chapter 2)

Tuesday, September 15: Two-state systems and spin 1/2 (Chapter 3)

Thursday, September 17: Linear algebra review, orthogonal and unitary groups (Chapter 4)

Tuesday, September 22: Lie algebras and Lie algebra representations (Chapter 5)

Thursday, September 24: Rotations and spin in 3 and 4 dimensions (Chapter 6)

Tuesday, September 29: The spin 1/2 particle in a magnetic field (Chapter 7)

Thursday, October 1: Representations of SU(2) and SO(3) (Chapter 8)

Tuesday, October 6 : Tensor products (Chapter 9)

Thursday, October 8: Review

Tuesday, October 13: Midterm exam (material through Chapter 9 of the notes)

Thursday, October 15: Momentum and the free particle (Chapter 10)

Tuesday, October 20: Fourier analysis and the free particle (Chapter 11)

Thursday, October 22: Position and the free particle (Chapter 12)

Tuesday, October 27: The Heisenberg group and the Schr\"odinger representation (Chapter 13)

Thursday, October 29: The Poisson bracket and symplectic geometry (Chapter 14)

Thursday, Novermber 5: Hamiltonian vector fields and the moment map (Chapter 15)

Tuesday, November 10: Quadratic polynomials and the symplectic group (Chapter 16)

Thursday, November 12: Quantization (Chapter 17)

Tuesday, November 17: Semi-direct products (Chapter 18)

Thursday, November 19: Quantum free particles and representations of the Euclidean group (Chapter 19)

Tuesday, November 24: Representations of semi-direct products (Chapter 20)

Tuesday, December 1: Central potentials and the hydrogen atom (Chapter 21)

Thursday, December 3: The harmonic oscillator (Chapter 22)

Tuesday, December 8: Coherent states (Chapter 23)

Thursday, December 10: Review


Other Textbooks

A standard physics textbook at the upper-undergraduate to beginning graduate level should be available to consult for more details about the physics and some of the calculations we will be studying.  A good choice for this is

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 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:

Naive Lie Theory, John Stillwell
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

Online Resources

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


Previous Courses

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