University of Illinois at Urbana-Champaign · Department of Physics

Physics 581

Quantum Mechanics II

Academic Year 2006/2007

Spring Semester 2007

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: 217-244-7704

Time: 1:00 pm - 2:20 pm, Monday/Wednesday
Place: Rm. 144 Loomis Laboratory
CRN: 36784
Credit: 1 unit.
Office Hours: Tuesdays 4:00-5:00 pm Rm 2119 ESB.

TA: Dimitrios Galanakis
Office: 4107 ESB
Phone: 333-5137
Office Hours: Thursdays 3:00-4:00 pm Rm 4107 ESB

The Department of Physics offers a two-semester long sequence of graduate level Quantum Mechanics, Physics 580 and Physics 581. Both courses are essentially thought of as a single course, with Physics 580 covering the basic material and Physics 581 the more advanced topics. This is the continuation of the year long course on Quantum Mechanics. I taught part I in the Fall Semester, 2006. I will assume that you are familiar with what I taught in Physics 580.


Course Plan

Time Evolution of Observables and Time Dependent perturbation Theory

Heisenberg and Schrödinger pictures. The evolution operator. Time ordered products. Time-dependent perturbation theory. Fermi's Golden Rule. Applications.

Path Integrals and Quantum Mechanics

Path integrals and the Superposition Principle. The Feynman Path Integral. Path integrals in imaginary time and Statistical Mechanics. Calculus of path integrals for harmonic systems. Functional determinants. The semi-classical approximation. The double well and tunneling.


The group of rotations and its representations. Integer and half integer representations. Rotations in spin space. Spin wave functions. Spin Hamiltonians. Spin-Orbit Interaction. Spin Precession and Spin Resonance. Adiabatic processes and Berry phases.

Theory of Angular Momentum and the Rotation Group in Quantum Mechanics

Addition of angular momenta. Clebsch-Gordan coefficients. Clebsch-Gordan methods. Applications. Scattering of distinguishable particles with spin. Rotation and tensor operators. The Wigner-Eckart Theorem and applications.

Identical Particles, Multi-Particle Systems and Second Quantization

Quantum statistics and identical particles. The EPR paradox, "reality" and Bell's inequalities. Spin-Statistics connection. Multiparticle states. Symmetrization and antisymmetrization. Young tableaux. Exchange interactions and spin. Scattering of identical particles. Fermions and Bosons. Second Quantization. Construction of multiparticle states with the proper symmetry. Fock space. Creation and anihilation operators. Normal ordering. Hamiltonians. Free fermions and the filled Fermi sea. Interacting fermions. Multi-electron atoms and simple molecules. Hartree and Hartree-Fock approximations. Coulomb interactions. Quantum matter at finite density and collapsed stars.

Interaction of Radiation with Matter

Review of Classical Electromagnetism. Vector potentials. Maxwell's Equations. Energy and momentum of the electromagnetic field. Classical interaction of matter with the electromegnetic field. Minimal coupling and gauge invariance. The Schrödinger equation for matter coupled to the electromagnetic field. Absorption and emission of light.

The Quantized Electromagnetic Field

Coulomb gauge. Transverse photons. Creation and anihilation operators. Mode expansions. Momentum, energy and angular momentum. Interaction of non-relativistic particles with the quantized electromagnetic field. Spontaneous emission. Einstein's coefficients. Electric dipole transitions. Cross sections and Sum Rules.

Relativistic Quantum Mechanics

Quantum mechanics and Special Relativity. The Klein-Gordon Equation. Solutions and particle interpreation. The Klein-Gordon equation and scalr field theory. The Dirac Equation. Non-relativistic limit. Relativistic covariance of the Klein-Gordon and Dirac equations. Particles and Antiparticles. The Dirac sea. The Spin-Statistics Theorem. The Klein paradox and pair production. The Hydrogen atom and the Dirac equation; bound state spectrum.

Syllabus for Quantum Mechanics I, Physics 580, Fall Semester 2006


There will be a total of six (6) homework sets; the final set will play the role of the (take home) final exam. The final exam will have the same weight as any of the other homework sets. However, since it is the Final Exam you must take it or otherwise you will not be able to pass this class. The homework sets are due on the due date posted on this webpage and have to be deposited at the Physics 581 Homework Box (at the Loomis end of the Loomis-MRL Interpass, second floor), 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 581 Gradebook


Homework Set No. 1 ; pdf file

posted on Wednesday 1/17/2007; Due date Monday 2/5/2007

Solutions to Homework Set No. 1

Homework Set No. 2 ; pdf file

posted on February 4, 2007; Due date Monday February 19, 2007

Solutions to Homework Set No. 2

Homework set No. 3 ; pdf file

posted on Wednesday February 28, 2007; Due date Wednesday March 14, 2007

Solutions to Homework Set No. 3

Homework set No. 4 ; pdf file

posted on Friday March 23, 2007; edited on Friday March 30; Due date Monday April 9, 2007

Solutions to Homework Set No. 4

Homework set No. 5 ; pdf file

posted on Monday April 16, 2007; Edited on Tuesday April 24, 2007; Due date Monday April 30, 2007

Solutions to Homework Set No. 5

Homework set No. 6/ Final Exam ; pdf file

posted on Sunday April 29, 2007; Due date Friday May 5, 2007, at 7:00 pm


Required textbooks

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).

Kurt Gottfried and Tung-Mow Yan, "Quantum Mechanics: Fundamentals", 2nd. edition, Springer (2003).

Ernest S. Abers, "Quantum Mechanics", Pearson/ Prentice Hall (2004).

J. J. Sakurai, "Modern Quantum Mechanics", Revised Edition,Adison-Wesley/Longman (1994).

Last updated 5/5/2007