Department of Physics
University of Illinois at UrbanaChampaign
Room 2119 ESB, MC704,
1110 W Green St, Urbana, IL 618013080
Phone: 2173334409
Fax: 2172447704
Email efradkin@illinois.edu
http://eduardo.physics.illinois.edu/homepage/
Time: 2:003:20 pm Mondays/Wednesdays
Place: Rm. 144 Loomis
CRN: 30709
Credit: 1 unit.
Office Hours: Tuesdays 4:00pm 5:00 pm, Rm 2119 ESB
TA: Simon Lin
Email: shanlin3@illinois.edu
Office Hours: Fridays 34 pm, Loomis Interaction Room (2nd floor, LLP)
TA: Min LI
Email: minl2@illinois.edu
Office Hours: Thursdays 23 pm, Loomis Interaction Room (2nd floor, LLP)
The Department of Physics offers a twosemester 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. In will teach the advanced
graduate level of Quantum Mechanics, Physics 581, in the Spring semester of the
Academic year 2017/2018. For Physics 580 I will assume that the students are
familiar with Quantum Mechanics at the level of an undergraduate course of the type offered
here. I will also assume that the students are familiar with
the Lagrangian and Hamiltonian descriptions of Classical
Mechanics, say at the level of Landau and Lifshitz and/or Goldstein.
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, as well as with standard material in partial differential
equations. Much of the necessary background is at the level covered in the MMA and MMB courses
offered here. Students are urged to review this background
material before coming to class. I prepared a set of notes on this material.
Below you will find a link to these notes.
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 selfadjoint operators. Eigenvalues and
eigenvectors. Finite dimensional and infinite dimensional spaces. This is a review of material that you should know from your undergraduate math classes.
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
part 1
part 2
part 3
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 AharonovBohm effect. The integer quantum Hall effect.
Symmetries in Quantum Mechanics
Symmetry transformations. Transformation groups. Point groups. Continuous groups
and the theory of angular momentum. Representations. Irreducible
representations.
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.
Scattering Theory
Scattering processes in Classical and Quantum Mechanics. Green functions. Cross
sections. Timeindependent perturbations. Born approximations
Perturbation theory
Time independent perturbations. RayleighSchrödinger and BrillouinWigner expansions.
The role of conservation laws. Degenerate states. Applications.
The semiclassical limit
Wave functions and the semiclassical limit. Wave packets. BohrSommerfeld quantization
rules. The WentzelKramersBrillouin approximation. Tunneling.
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 semiclassical
approximation. The double well and tunneling.
Syllabus for Quantum Mechanics II,
Physics 581, Spring Semester 2018
Grades
There will be a total of seven (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 580 Gradebook
Homeworks
Homework Set No. 1 ;
pdf file
posted Sunday July 30, 2017; Due date: Wednesday, September 20, 2017, 5:00 pm
Solutions to Homework Set No. 1
Homework Set No. 2 ;
pdf file
posted Wednesday September 20, 2017; Due date: Friday October 6, 2017, 5:00 pm
Solutions to Homework Set No. 2
Homework set No. 3 ;
pdf file
posted Wednesday October 4, 2017;Due date: Friday October 20, 2017, 5:00 pm
Solutions to Homework Set No. 3
Homework set No. 4 ;
pdf file
posted Friday October 20, 2017; Due date: Friday November 3, 2017 , edited version posted on October 30
Solutions to Homework Set No. 4
Homework set No. 5 pdf file
posted Monday November 6, 2017; Due date: Friday November 17, 2017
Solutions to Homework Set No. 5
Homework set No. 6; pdf file
posted Friday November 17, 2017; Due date: Friday December 8, 2017, 5:00 pm
Solutions to Homework Set No. 6
Homework set No. 7/ Take Home Final Exam; pdf file
posted Saturday December 9, 2017 ; Due date: Sunday December 17, 2017, 5:00 pm
Note: This is a takehome Final Exam. The solutions must send me by email (to efradkin@illinois.edu) the pdf file with your solutions before Sunday 12/17/2017 at 5:0 0pm.
You should not give the solutions to the TA nor put them in the Physics 580 homework box.
No late sets 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.
Bibliography
Required textbooks
R. Shankar, "Principles of Quantum Mechanics", Second Edition, Plenum Press
(1994).
J. J. Sakurai, "Modern Quantum Mechanics", Revised Edition, AddisonWesley Longman (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, McGrawHill (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).
