Introduction to Quantum Mechanics: Mathematics W4392 (spring 2013)


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

This course will be a continuation of the fall 2012 course Mathematics W4391 which covered some of the basic concepts and examples of quantum mechanics, emphasizing the role of symmetry, Lie groups and their unitary representations.  Students who have taken a basic quantum mechanics course and have knowledge of groups and representations should be able to follow the spring course.

Topics covered will include some of the conventional material of a physics advanced quantum mechanics course (relativistic quantum mechanics, quantization of the electromagnetic field, path integrals, field theoretical methods), as well as some more mathematical topics (anti-commuting variables and Clifford algebras, representations of the Poincare group).

Tentative Syllabus

Lecture Notes

Lecture notes from the fall semester are available here.

Since the perspective of the course will be different than any of the available textbooks, I plan to produce some lecture notes, which will appear periodically here.

The Fermionic oscillator
Weyl and Clifford algebras
Clifford algebras and geometry
Anti-commuting variables and pseudo-classical mechanics
Spinors (highly preliminary)
Some simple examples of supersymmetry

Clifford algebras and the fermionic oscillator(under revision)

Lagrangian methods, the path integral
Non-relativistic quantum field theory
Second quantization and quantum fields
Minkowski space and the Lorentz group
Representations of the Lorentz group
Semi-direct products and their representations
The Poincare group and its representations
The Klein-Gordon equation and scalar quantum fields
The Dirac equation and spin-1/2 field
U(1) gauge symmetry and coupling to the electromagnetic field
Quantization of the electromagnetic field: the photon
An introduction to the standard model

Problem Sets

There will be problem sets due roughly every couple weeks, as well as a final project.  The last part of the class will likely consist of presentations by students on topics chosen in consultation with the instructor.

Problem Set 7, due Wednesday February 20.
Problem Set 8, due Monday March 25.
Problem Set 9, due Monday April 15.

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:

Quantum Field Theory: A tourist guide for Mathematicians, by Gerald Folland.  AMS, 2008
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.