Physics 480

Quantum Mechanics I

Academic Year 2002/2003

Spring Semester 2003

Instructor: Professor Eduardo Fradkin

Department of Physics
University of Illinois at Urbana-Champaign
Room 2119 ESB, MC-704,
1110 W Green St, Urbana, IL 61801-3080
Phone: 217-333-4409
Fax: 217-333-9819
E-mail efradkin@uiuc.edu
http://w3.physics.uiuc.edu/~efradkin/

Time: 1:00-2:20 pm Tuesdays/Thursdays
Place: Rm. 158 Loomis
Call Number: 06569
Credit: 1 unit.
Office Hours: Wednesdays 4:00pm-5:00pm, Rm 301B LLP

TA: Mr. Kalin Vetsigian
Office: Room 3105 ESB
Phone: 333-6311
E-mail: vetsigia@uiuc.edu
Office Hours:

The Department of Physics offers a two-semester long sequence of graduate level Quantum Mechanics, Physics 480 and Physics 481. Both courses are essentially thought of as a single course, with Physics 480 covering the basic material and Physics 481 the more advanced topics. In will teach the advanced graduate level of Quantum Mechanics, Physics 481, in the Fall semester of the Academic year 2003/2004. For Physics 480 I will assume that the students are familiar with Quantum Mechanics at the level of an undergraduate course, as well as with the Lagrangian and Hamiltonian descriptions of Classical Mechanics. At the beginning of the courses I will present a quick review of background material needed to follow the class. However I will assume that the students are familiar with the fundamental concepts of linear algebra, vector spaces, and calculus on functions of a complex variable, in particular with the methods of contour integration and residues. Students are urged to review this background material before coming to class.

Announcements

Course Plan

Mathematical Background

Review of the mathematics underlying Quantum Mechanics. Linear vector spaces. Inner product. Dual spaces. Dirac notation. Subspaces. Linear operators. Linear transformations. Hermitean and self-adjoint operators. Eigenvalues and eigenvectors. Finite dimensional and infinite dimensional spaces.

Review of Classical Mechanics

The Lagrangian and the Action. The Least Action Principle. Hamiltonian formulation of Classical Mechanics. Poisson brackets. Relation between symmetries and conservation laws in Classical Mechanics.

The Postulates of Quantum Mechanics

What is wrong with Classical Mechanics?. Blackbody radiation and the Planck spectrum. The Photoelectric effect. Atomic spectra. Double slit experiments. Particles and waves. Photon polarization experiments.

Quantum states. Measurements and wave functions. Operators and physical observables. The Uncertainty Principle. The Superposition Principle. The Correspondence Principle and the classical limit. Probabilistic interpretation of Quantum Mechanics. Energy and Momentum. The Hamiltonian operator. Stationary states. The Heisenberg representation of operators. The density matrix. Momentum. Uncertainty Relations.

The Schrödinger Equation

The Schrödinger equation. Properties. Probability current. Stationary states. Quantum mechanical motion in one dimension. The potential well. The linear harmonic oscillator. Uniform field. Transmission coefficient. Motion in a magnetic field in two dimensions. The Aharonov-Bohm effect. The integer quantum Hall effect.

Angular Momentum

Eigenvalues and eigenvector of the angular momentum. Matrix elements. Parity. Addition of angular momenta.

Motion in a Central Field

Spherical waves. The hydrogen atom. Partial wave decomposition of a plane wave. Motion in a Coulomb field. Bound states and scattering states.

Perturbation theory

Time independent perurbations. Rayleigh-Schrödinger and Brillouin-Wigner expansions. The role of conservation laws. Degenerate states. Applications.

Scattering Theory

Scattering processes in Claasical and Quantum Mechanics. Green functions. Cross sections. Time-independent perturbations. Born approximations

The semi-classical limit

Wave functions and the semi-classical limit. Wave packets. Bohr-Sommerfeld quantization rules. The Wentzel-Kramers-Brillouin approximation. Tunneling.

Symmetries in Quantum mechanics

Symmetry transformations. Transformation groups. Point groups. Continuous groups and the theory of angular momenta. representations. Irreducible representations.

Syllabus for Quantum Mechanics II, Physics 481, Fall Semester 2003

Grades

There will be a total of six (7) homework sets. The last set will be your final exam. The final exam will weigh 1/7 of the grade and the homework sets the remaining 6/7. The homework sets are due on the due date posted on this webpage and have to be deposited at the TA's mailbox before midnight on that date. There will be a 20% grade penalty on late homework sets and no homework sets will be accepted of they are more than two days late (barring extenuating circumstances which will only be considered by the Instructor, not by the TA).

You may check your grades by looking at your entry in the Physics 480 Gradebook

Homeworks

Homework Set No. 1 ; pdf file

posted on 1/28/2003; Due date 2/7/2003

Solutions to Homework Set No. 1

Homework Set No. 2 ; pdf file

posted on 2/7/2003; Due date 2/17/2003

Solutions to Homework Set No. 2

Homework set No. 3 ; pdf file

posted on 2/20/2003; Due date 2/28/2003

Solutions to Homework Set No. 3

Homework set No. 4 ; pdf file

posted on 3/10/2003; Due date 3/21/2003

Solutions to Homework Set No. 4

Homework set No. 5 pdf file

posted on 4/1/2003; Due date 4/12/2003

Solutions to Homework Set No. 5

Homework set No. 6; pdf file

posted on 4/15/2003; Due date 4/25/2003

Solutions to Homework Set No. 6

Homework set No. 7/ Final Exam; pdf file

posted on 4/25/2003; Due date 5/11/2003

Bibliography

Required textbooks

Note: I had originally assigned the book "Quantum Mechanics" by Landau and Lifshitz as a textbook. Unfortunately it will not be available until well into the semester.

R. Shankar, "Principles of Quantum Mechanics", Second Edition, Plenum Press (1994).

Gordon Baym, "Lectures on Quantum Mechanics", Addison Wesley (1990).

Recommended textbooks

L. D. Landau and E. M. Lifshitz, "Quantum Physics", Third Edition , Course of Theoretical Physics, Volume 3. Pergamon Press (1991).

Eugene Merzbacher, "Quantum Mechanics", Second Edition, J. Wiley & Sons (1970).

Leonard Schiff, "Quantum Mechanics", Third Edition, McGraw-Hill (1968).

Albert Messiah, "Quantum Mechanics", Dover (1999).

Paul A. M. Dirac, "The Principles of Quantum Mechanics", Oxford Science Publications, Fourth Edition (1958).

Richard P. Feynman, Robert B. Leighton and Matthew Sands, "The Feynman Lectures on Physics", Volume 3, Addison Wesley (1969).