Physics 718

Graduate Quantum Mechanics II

Spring 2005

Instructor: Lev Kaplan

Lectures: MWF 10:00 - 10:50, in ***

Textbooks: Messiah, Quantum Mechanics

Dirac, Principles of Quantum Mechanics, 4th edition

Possibly useful reference books: Sakurai, Modern Quantum Mechanics (especially early chapters)

Cohen-Tannoudji et al., Quantum Mechanics (comprehensive treatment)

Landau et al., Quantum Mechanics

Merzbacher, Quantum Mechanics (standard "traditional” text)

Shankar, Principles of Quantum Mechanics (introductory level)

Ballentine, Quantum Mechanics: A Modern Development (conceptual)

Office: 5046 Percival Stern Hall

Office hours: M 1:00 – 2:00 and Tu 2:00 – 3:00 (to be confirmed!), or by appointment

Email: lkaplan@tulane.edu [A great way to ask a question or arrange an appointment]

Telephone: 504-862-3176 (x3176) [Please leave a message if I’m not there or try email]

Welcome to Graduate Quantum Mechanics II at Tulane University!

General Course Objectives and Requirements:

This course, the second half of a two-semester sequence, builds on the basic principles and mathematical formalism introduced in the first semester, and is intended to introduce you to several important classes of problems and techniques in quantum mechanics. The emphasis will be on fundamentals and general methods rather than on detailed applications, with specific examples being used primarily to illustrate the general development. As before, our focus will be on the physics rather than on mathematical rigor. Topics that may be covered include: variational methods, interaction picture and time-dependent perturbation theory, Fermi’s golden rule, scattering theory and Born approximation, discrete and continuous symmetries, identical particles and exchange symmetry, central potentials, addition of angular momentum, semiclassical propagator and Feynman path integral, Green’s functions and resonances, occupation number representation, gauge transformations and Aharonov-Bohm effect.

Grading:

Homework: 30%

Midterm exam: 30%

Comprehensive final exam: 40%

Homework:

Homework will be assigned every one or two weeks. The assignments may include derivations and examples that extend the class discussion. Doing the homework in a timely manner will enable you to keep up with the material and be prepared when exam time arrives.

Examinations:

One midterm (March 4) and a comprehensive final exam (May 5) will compose 70% of your final grade. The date of the midterm is subject to minor adjustment, but the date of the comprehensive final exam is set by the university. Each exam may contain a combination of definition/conceptual questions (short answer), derivations, and calculations. Exam problems and questions will be drawn from material covered in class as well as from homework problems. More information will be provided prior to the date of each exam.

Tentative Course Outline:

This outline is subject to likely adjustments, which will be announced in class.

1. Variational method (Messiah XVIII)

Ground state and excited states

2. Interaction picture and time-dependent perturbation theory (Messiah XVII)

Interaction (intermediate) picture, evolution of states and operators

Example: 2-level systems, Rabi oscillations, resonance

Dyson series, perturbation theory, transition probability

Constant perturbation, harmonic perturbation

Transitions into quasi-continuum, Fermi’s golden rule

Sudden and adiabatic perturbations

3. Symmetries (Messiah XV)

Review: symmetries and conservation laws, momentum, angular momentum, energy

Discrete and conitnuous symmetries, symmetry and degeneracy

Examples: parity, rotation, lattice translation (Bloch’s theorem)

Time reversal, antiunitary opertaors

4. Systems of identical particles (Messiah XIV)

Permutation symmetry, Symmetrization postulate

Bose-Einstein and Pauli-Dirac statistics, Pauli exclusion principle, exchange density

5. Central potentials and hydrogen atom (Messiah XI)

Schrodinger equation in spherical coordinates, spectrum, degeneracy

Eigenfunctions and spherical harmonics

Two-body systems, reduced mass

6. Addition of angular momentum (Messiah XIII)

Addition theorem, eigenvalues and eigenvectors

Clebsch-Gordon coefficients

Example: spin-½ particles

7. Scattering (Messiah X)

Scattering amplitude, Lippmann-Schwinger equation

Differential and total cross sections

Wave packet description

Born approximation

Optical theorem

Partial waves and phase shifts

8. Green’s function and resolvent

Time and energy domains, bound states and resonances

9. Feynman formulation

Van-Vleck propagator, connection with classical mechanics

Feynman’s path integral

10. Gauge transformations

Electromagnetism, canonical and mechanical momentum

Aharnov-Bohm effect

Magnetic monopoles and electric charge

11. Occupation number representation (Dirac VI, Messiah XIII)

Bosons and fermions

Operators in second quantization

About me:

I am an Assistant Professor at Tulane, and prior to coming here I did research at the University of Washington in Seattle. I was born in Latvia, went to school in New Jersey, did my undergraduate studies at the University of Pennsylvania, and my graduate “work” at Harvard University (on particle theory). My present research interests center around quantum chaos, the quantum mechanics of “generic” systems. I strongly encourage you to stop by my office to talk about physics, the department, Tulane, questions, concerns, suggestions, or anything else that may be on your mind.