Physics 717

Graduate Quantum Mechanics I

Fall 2004

Instructor: Lev Kaplan

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

Textbooks: Dirac, Principles of Quantum Mechanics, 4th edition

Messiah, Quantum Mechanics

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

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

Landau et al., Quantum Mechanics

Merzbacher, Quantum Mechanics (traditional "standard" text)

Shankar, Principles of Quantum Mechanics (introductory level)

Ballentine, Quantum Mechanics: A Modern Development (conceptual)

Office: 5046 Percival Stern Hall

Office hours: to be arranged after first class meeting, 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 I at Tulane University!

General Course Objectives and Requirements:

This course, part one of a two-semester sequence, is intended to help you master the basic principles and mathematical formalism of quantum mechanics at the introductory graduate level. The emphasis will be on fundamentals and general structure of the theory rather than on detailed applications, with specific examples used primarily to illustrate the general development. Although the course is self-contained, a solid background in quantum mechanics at the undergraduate level is strongly recommended. The required mathematical tools (e.g., linear algebra) will be introduced as needed. While some formal aspects of quantum mechanics will be emphasized, the focus will be on the physics rather than on mathematical rigor. Topics covered will include states and operators, measurement theory, n-level and 2-level systems, representations, unitary transformations, Schrodinger and Heisenberg pictures, quantum statistical mechanics, angular momentum, time-independent and time-dependent perturbation theory, and many-particle systems. The classical limit of quantum mechanics will be discussed where appropriate.

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.

Examinations:

One midterm and a comprehensive final exam will compose 70% of your final grade. The date of the midterm is subject to very 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 Schedule:

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

Brief introduction/history (Messiah I, II)

Failures of classical mechanics, old quantum theory

Wave mechanics and matrix mechanics

Quantum "kinematics" (Messiah VII)

States and vectors, complex vector space structure, superposition principle

Kets and bras, scalar product, norms, overlaps, probabilities

Linearly dependent and independent states, complete orthonormal sets

N-state systems, connection with matrix mechanics

Example: 2-state systems ("qubits")

Tensor products

Operators and observables (Messiah V, VII, VIII)

Linear operators, adjoint, self-adjoint, matrices, projection operators

Eigenvalues and eigenstates, definition of observable, completeness relation

Discrete and continuous spectra, normalization, delta function, normalizability

Functions of observables

Example: particle in one or dimensions

Expectation value and fluctuations

Example: 2-state systems (spin 1/2), Pauli matrices

Representations and unitary transformations (Messiah VII, VIII, V)

Example: position representation

Change of representation, unitary operators, effect on states and observables, norm conservation

Spin 1/2 systems

Momentum representation and wave functions, commutation relations, Fourier transforms

Uncertainty principle and displacement operators (Messiah IV, V, VII, VIII)

Commutation relations and uncertainty, position and momentum

Infinitesimal transformations, translations, generators, finite displacement operators

Complete sets of commuting observables

Time evolution ("dynamics") (Messiah VIII, VI)

Hamiltonian operator as generator, Schrodinger equation

Stationary states, stationary state basis

Finite time evolution and time ordering

Schrodinger, Heisenberg, intermediate pictures

Ehrenfest theorem and classical limit

Quantum commutators and classical Poisson brackets

Analogy with classical Hamilton's equations, canonical quantization

Constants of motion and symmetries

Example: particle in one or more dimensions

Gaussian wave packets

Energy-time uncertainty relation

Mixed states and statistical mechanics (Messiah VIII)

Pure and mixed states, density operator, traces, expectation values, entropy

Example: matrix mechanics

Time evolution of density operator

Thermal equilibrium and partition function

Example: Harmonic oscillator (Messiah XII)

Creation and annihilation operators, number operator, spectrum, expectation values

Thermal equilibrium

Position representation: wave functions

Wave packets and semiclassical description

Multi-dimensional or coupled harmonic oscillators

Example: angular momentum (Messiah XIII)

Rotations, commutation relations

Eigenstates of angular momentum

Special case: spin 1/2

Perturbation theory: time independent (Messiah XVI)

Degenerate perturbation theory

Perturbation theory: time dependent (Messiah XVII)

About me:

I am a first-year Assistant Professor at Tulane, and prior to coming here I did research at the University of Washington in Seattle. I was born in Latvia (former USSR), went to school in New Jersey, did my undergraduate studies at the University of Pennsylvania, and my graduate “work” at Harvard University (on the theory of high-energy particles). My present research interests include something called “quantum chaos.” If you want to know what that means, just ask. I strongly encourage you to stop by my office to talk about physics, Tulane, questions, concerns, suggestions, or anything else that may be on your mind.