University of Illinois at Urbana-Champaign · Department of Physics

Physics 581

Quantum Mechanics II

Academic Year 2022/2023

Spring Semester 2023

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: Tuesdays and Thursdays 2:00 pm - 3:20 pm
Place: Rm 276 Loomis Lab.
CRN: 36784
Credit: 1 unit.
Office Hours: Wednesdays 2:30 pm - 3:30 pm (by zoom) (registered students were given the link in an email)

TA: Mr. Zejun Liu
Office: TBA
Office Hours: Thursdays 4:00 pm to 5:00 pm (by zoom) (registered students were given the link in an email)

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. Unfortunately I did not teach part I (Physics 580) in the Fall Semester 2022 and this may cause some problems. Nevertheless I will have to assume that you are familiar with the standard syllabus for Physics 580 which I last taught in the Fall 2017. You can fund the syllabus for the Physics 580 course that I taught in the Fall 2017 (and pdf files of my handwritten lecture notes here .


Updated on Friday May 5, 2023

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. Coherent state path integrals and the path integral for spin

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. Quantum entanglement. Spin-Statistics connection. Multi-particle states. Symmetrization and antisymmetrization. Young tableaux. Exchange interactions and spin. Scattering of identical particles. Fermions and Bosons. Second Quantization. Construction of multi-particle states with the proper symmetry. Fock space. Creation and annihilation 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 electromagnetic 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 annihilation 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 interpretation. The Klein-Gordon equation and scalar 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 2017


Please find below the HW sets. Please notice the clearly shown deadlines. There will be a total of 7 (7) homework sets. The final set will play the role of the (take home) final exam. The final exam will have the same weight (1/7) 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. To upload the pdf file of your work you will go to the Physics 581 space in my.physics. There you will find the link "Course uploads" and clicking on it you will be able to upload your solutions and download the graded solutions. There you will see a menu with the list of homeworks for Physics 581 and you will choose which homework solution you wish to upload. Alternatively, you can go to the link to access the relevant homework folder. After the homework is graded by the TA you will be able to access and download the graded work by using this same link. The homework sets are due on the due date posted on this webpage and have to be uploaded 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 three 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 Friday January 20, -203; Due date Friday February 3, 2023, at 9:00 pm US Central Standard Time

Solutions to Homework Set No. 1

Homework Set No. 2 ; pdf file

posted Friday February 3, 2023; Due date Friday February 17, 2023, 9:00 pm US Central Standard Time

Solutions to Homework Set No. 2

Homework set No. 3 ; pdf file

posted Friday February 17, 2023; Due date Friday March 3, 2023, 9:00 pm US Central Standard Time ; deadline extended to Friday March 10, 9 pm US CST.

Solutions to Homework Set No. 3

Homework set No. 4 ; pdf file

posted Friday March 10, 2023; Due date Sunday March 26, 2023, 9:00 pm US DST

Solutions to Homework Set No. 4

Homework set No. 5 pdf file

posted Friday March 24, 2023; Due date Sunday, April 9, 2023, 9:00 pm US CDT

Solutions to Homework Set No. 5

Homework set No. 6/; pdf file

posted Monday April 10, 12 pm US CDT (edited on 4/19/23) Due date Friday April 28, 2023, 9:00 pm US CDT

Solutions to Homework Set No. 6

Homework set No. 7 / Take Home Final Exam; pdf file

posted Friday April 28, 2023; Due date: Thursday May 11, 2023, 9:00 pm US CDT

Note: This is a take-home Final Exam. No late uploads will be accepted. Since this problem set is a take home Final Exam, you must turn in your solutions to pass this course. You must make sure that the pdf file is clearly legible!. Please either prepare the solutions in LaTeX . If your solutions are handwritten you must write them with dark pen with clearly legible handwriting in double spaced paper. I cannot grade solutions with bad handwriting and/or with random organization. They must be clearly organized with your results shown clearly. Your solutions must be uploaded in the same manner as the HW sets.


Required textbooks

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

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

Recommended textbooks

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

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

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).

Last updated 5/9/2023