Physics 580

Quantum Mechanics I

Academic Year 2017/2018

Fall Semester 2017/CENTER>

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
E-mail efradkin@illinois.edu
http://eduardo.physics.illinois.edu/homepage/

Time: 2:00-3:20 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
E-mail: shanlin3@illinois.edu
Office Hours: Fridays 3-4 pm, Loomis Interaction Room (2nd floor, LLP)

TA: Min LI
E-mail: minl2@illinois.edu
Office Hours: Thursdays 2-3 pm, Loomis Interaction Room (2nd floor, LLP)

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. 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.

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. 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 Aharonov-Bohm 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. Time-independent perturbations. Born approximations

Perturbation theory

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

The semi-classical limit

Wave functions and the semi-classical limit. Wave packets. Bohr-Sommerfeld quantization rules. The Wentzel-Kramers-Brillouin 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 semi-classical 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 take-home 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, Addison-Wesley 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, 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).