MAT 494/598 Special Topics: Mathematics of Quantum Mechanics

TITLE OF COURSE:                 Mathematics of Quantum Mechanics
INSTRUCTOR:                           Dr. Sergei K. Suslov
TIME:                                            3:00-4:15 pm, TuTh
LOCATION:                                 LSA109
LINE #:                                           


The main purpose of this course is to help beginners explore the World of Quantum Mechanics, one of the most important scientific discoveries of the last century, and a subject that is very important in the education of student majoring in science or engineering.

This course uses mathematical methods which are not always part of the usual course sequence. It is designed as an introduction to quantum  mechanics, with emphasis on the mathematical aspects of the theory. This includes the Schrodinger equation, the WKB approximation, angular momentum and spin, some applications of the theory of symmetry in quantum mechanics, motion in a magnetic field and, if time permits, the Dirac equation and/or other topics in relativistic theory.

All mathematical tools (aspects of analytic function theory, orthogonal polynomials,  special functions, group representations, and asymptotic methods) will be introduced as needed.

PREREQUISITES: MAT 272 and 342, or instructor's approval.

A. Das, Lectures on Quantum Mechanics, AMS & Hindustan Book Agency, 2003
A. Messiah, Quantum Mechanics, Dover, 2001
or any other similar textbook on non-relativistic quantum mechanics.

I. I. Gol'dman and V. D. Krivchenkov, Problems in Quantum Mechanics, Dover, New York, 1993;
S. Flugge, Practical Quantum Mechanics, Springer, 1999 (reprint of the 1994 edition);
L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Third revised edition 1977, reprinted by Butterworth-Heinemann, 1998;
E. Merzbacher, Quantum Mechanics, Wiley, New York, 1998;
A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhauser, Boston, 1988.

For more information contact Sergei Suslov:
office:    PSA 621
office hours: TTh 1:30-2:50 pm, or by appointment
phone # (480) 965-8987


1.   The WKB approximation. Bohr-Sommerfeld's quantization rule.
2.   Perturbation theory. Perturbations independent of time. The secular equation.
3.   Motion in a Coulomb field (sperical coordinates and parabolic coordinates).
4.   Rotational symmetry: the rotation operator, angular momentum and conservation laws.
5.   The representations of the rotation group.
6.   Addition of angular momenta. Clebsch-Gordan coefficients. Racah coefficients.
7.   Irreducible tensor operators.
8.   Hidden symmetry of the hydrogen atom.
9.   Scattering. The cross section. The Born approximation.
10.  Scattering in the Coulomb field.
11.  The spin. The spin operator, Pauli matrices and spin angular momentum.
12.  Motion in magnetic field. Schrodinger equation in a magnetic field.
13. Motion in a uniform magnetic field. Landau levels.
14. A hydrogen atom in an electric field. The Stark effect.
15.  The Dirac equation: solutions of the free field Dirac equation.
16.  The Dirac equation: central forces and the hydrogen atom.
17.  Tree method for multidamensional Laplace equation.


Gol'dman and Krivchenkov:

Section 1: # 1, 2, 3, 4, 5, 6, 7, 10, 11, 12

Section 3: # 1, 2, 3

An Introduction to Special Functions

Lecture Notes  ps pdf

JMP Paper on Dirac's Equation

"The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems," by S.K. Suslov and B.Trey, Journal of Mathematical Physics (Vol.49, Issue 1):

DOI: 10.1063/1.2830804

Section 6 contains solution of the wave equation of Dirac for the Coulomb
Some other recent QM  papers:

"The Propagator of a Charged Particle with a Spin in a Uniform Magnetic and a Perpendicular Electric Fields"
by Ricardo Cordero-Soto, Raquel Lopez, Erwin Suazo, and Sergei Suslov

"The Cauchy Problem for a Forced Harmonic Oscillator" by Raquel Lopez and Sergei Suslov

"Solution of the Cauchy Problem for a Time-Dependent Schoedinger Equation" by Maria Meiler,
Ricardo Cordero-Soto, and Sergei K. Suslov

Presentations: Wednesday, May XX,