Introduction to Quantum Mechanics: Mathematics W4391 (fall 2012)


Monday and Wednesday 1:10-2:25pm
Mathematics 307

This course will be an introduction to the subject of quantum mechanics, from a perspective emphasizing the role of symmetry.  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.  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.

Tentative Syllabus

Lecture Notes

The lecture notes for the course have been combined into one document:

Fall course lecture notes

This document will continue to be edited and improved, the chapter versions below will not be updated and at some point will be removed.

Since the perspective of the course will be different than any of the available textbooks, especially at the beginning, I plan to produce some lecture notes, which will be available here.  There will also be copies of readings from various sources that will be made available to supplement the lecture notes and the course textbook.

Introduction and overview
U(1) and charge
Two-state quantum qystems: SU(2) and spin 1/2
Review of linear Algebra, orthogonal and unitary Groups
Lie algebras and Lie algebra representations
The rotation and spin groups in 3 and 4 dimensions
The spinor representation, the spin 1/2 particle in a magnetic field
Representations of SU(2) and SO(3)
Tensor products, entanglement, and addition of spin
Energy, momentum and the quantum free non-relativistic particle
The Heisenberg group and the Schrodinger representation
Poisson brackets and quantization
The harmonic oscillator
Angular momentum and central potentials

Problem Sets

There will be problem sets due roughly every other week, a midterm (October 17) and a final exam.

Problem Set 1 (Due September 24)
Problem Set 2 (Due October 8)
Problem Set 3 (Due October 22)
Problem Set 4 (Due November 7)
Problem Set 5 (Due November 29)
Problem Set 6 (Due December 12)

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.

The book

Linearity, Symmetry and Prediction in the Hydrogen Atom, Stephanie Singer, Springer, 2005.

covers some of the material we will cover, especially the hydrogen atom spectrum calculation, from a point of view similar to the one of this course.  It is available free on SpringerLink from Columbia addresses at this URL.  At this site one can also purchase a printed copy of the book for $24.95

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.

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)