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PY 511-512 -- QUANTUM MECHANICS

Instructor: Kenneth Lane, PRB 579, ext. 3-4512; .
Class: Tuesday and Thursday, 11:00 AM-12:30PM in PRB 261.
Discussion: Wednesday, 10:00-11:00 AM in PRB 146.
Office hours: Monday, 3:00-5:00 PM or by appointment.
Attendance at class and discussion section is strongly recommended.
Grader: Tonguç Rador, PRB 565, ext. 3-6065;

Principal Texts:
Quantum Mechanics, by C. Cohen-Tannoudji, B. Diu and F. Laloë;
Feynman Lectures on Physics, Vol. III, by R. P. Feynman, R. B. Leighton
and M. Sands;
Notes by Kenneth Lane.

Supplemental Texts (on reserve):
Quantum Mechanics, Vols. I and II, by A. Messiah
Quantum Mechanics, Vol. I, by K. Gottfried
Quantum Mechanics - Nonrelativistic Theory, by. L. D. Landau and E. M. Lifschitz
The Principles of Quantum Mechanics, by P. A. M. Dirac
Introductory Quantum Mechanics, a good undergraduate text by R. L. Liboff
Lectures on Quantum Mechanics, by G. Baym
Quantum Mechanics, by E. Merzbacher
Quantum Mechanics, by L. Schiff

Your course grade is based on homework ( $30\%$ ), a take-home midterm exam ( $30\%$ ) and a take-home final exam ( $40\%$ ). Homework will be due in class on the first Tuesday following completion of lectures on the relevant chapter. You may work together on homework, but you should turn in your own work. Identical homeworks will not be graded; neither will late homework. You may NOT work together on exams.

PY 511-512 SYLLABUS -- Version 1

1.

Introduction / Review: Waves, Particles, and the Uncertainty Principle
Reading:
Cohen-Tannoudji, Chapter I; Feynman, Chapters 1-3, 16
Complements A_I-G_I

Lecture topics:

(a): Introductory comments.
(b): Quantum amplitudes.
(c): Uncertainty principle.
(d): Schrödinger equation; eigenfunctions; free wave packets.

2.

Mathematical Tools: State vectors, Dirac Notation, and Hilbert Space
Reading:
Cohen-Tannoudji, Chapter II)
Complements A_II-F_II

Lecture topics:

(a): State vectors (bras and kets); Hilbert space; superposition; inner product; orthonormal bases; completeness.
(b): Operators: unitary transformations and hermitian observables.
(c): Example: Momentum generates tranlations.
(d): Representations of states and operators.
(e): Complete sets of commuting observables (CSCOs).
Homework:
: Complement H_II - Problems 2,3,6,8,9,10; also do 11,12 for which solutions are provided.

3.

Basic Postulates of Quantum Mechanics
Reading:
Cohen-Tannoudji, Chapter III; Feynman, Chapters 3,7,8,20
Complements B_III, C_III; E_III-K_III

Lecture topics:

(a): Postulates and their physical interpretation.
(b): Schrödinger equation and its implications; probability current.
(c): Superposition principle.
(d): Generalized uncertainty principle.
(e): Evolution operator.
(f): Schrödinger and Heisenberg pictures.
(g): Ehrenfest's theorem and classical mechanics.
Homework:
: Complement L_III - Problems 3,5,8,9,14,15.

4.

One-Dimensional Potential Problems
Reading:
Cohen-Tannoudji, Complements H_I, J_I, M_III, N_III; O_III; Feynman, Chapter 13

Lecture topics:

(a)

Understanding terms in the Schrödinger equation (for fixed energy E):

i.: Kinetic and potential energy.
ii.: Wiggles and nodes in the wave function.
iii.: Classical turning point; behavior in classically forbidden regions.

(b)

One-dimensional potential barriers; reflection and transmission at a barrier; conservation of probability.

(c)

Tunneling through a barrier.

(d)

Bound states and boundary conditions: infinite square well; parity; finite well.

(e)

Transmission resonances; analyticity of the amplitude.

(f)

$\delta$ -function potential.

(g)

Periodic potential (if time permits).

Homework:

Complement K_I - Problems 2,3,6,7.

5.

Spin- $\frac{1}{2}$ and Two-State Systems
Reading:
Cohen-Tannoudji, Chapter IV; Feynman, Chapters 8-11)
Complements A_IV-C_IV, F_IV-H_IV

Lecture topics:

(a): Ammonia molecule: splitting of degenerate levels by a small (off-diagonal) ``perturbation''.
(b): General solution to the two-state problem for a time-independent H; Pauli matrices; diagonalization; perturbation theory; oscillation between unperturbed states.
(c): Spin- $\frac{1}{2}$ in static and dynamic magnetic fields.
Homework:
: Complement J_IV - Problems 1,2,5,7,8.

6.

Harmonic Oscillator Systems
Reading:
Cohen-Tannoudji, Chapter V
Complements A_V, B_V, E_V, F_V, H_V-K_V

Lecture topics:

(a): Operator solution of the problem.
(b): Wave functions (Hermite polynomials).
(c): An example to be chosen from the complements.
Homework:
: Complement M_V - Problems 2,5,6,7,8.

7.

Angular Momentum; Symmetry and Conservation Laws
Reading:
Cohen-Tannoudji, Chapters VI, IX, X; K. Lane notes; Messiah, Chapter 13; Feynman, Chapters 6, 17-18
Complements A_VI, B_VI, D_VI, E_VI; A_IX; D_IV, A_X-D_X

Lecture topics:

(a)

Orbital angular momentum and the fundamental commutation relations.

(b)

The spectrum of ${\vec J}^2$ and J_z.

(c)

Angular momentum generates rotations.

(d)

Symmetries and conservation laws.

(e)

Addition of angular momenta: Clebsh-Gordan series and coefficients.

(f)

Symmetries and group theory:

i.: Definitions of group, etc.
ii.: Lie groups: definition; generators; transformations.
iii.: Representation theory: definitions; products and sums.
iv.: SU(N) for $N = 2,3,\dots$
v.: Symmetry groups: $[\vec Q, \; H] = 0$ .
vi.: Fundamental theorem of symmetries in quantum mechanics: Degenerate states as (irreducible or reducible) representations.

(g)

Irreducible tensor operators and the Wigner-Eckart theorem.

(h)

Examples from particle physics.

Homework:

Following Chapter VI lectures, Complement F_VI - Problems 5,6,7,8.

Following group theory and Chapter X lectures, Complement B_IX - Problem 2;
Complement G_X - Problems 3,7,8 (or another problem to be determined).

8.

Bound States in the Coulomb and Other Central Potentials
Reading:
Cohen-Tannoudji, Chapter VII; Feynman, Chapter 19
Complements A_VII, B_VII, C_VII, D_VII

Lecture topics:

(a): The two-body problem with $V = V(\vec r_1 - \vec r_2)$ ;separation of c.m. and relative motions.
(b): Solution of the H-atom: spectrum and wave functions.
(c): Symmetry of the H-atom; understanding the energy degeneracies.
(d): Isotropic 3-dimensional harmonic oscillator and SU(3).
(e): H-atom in a uniform magnetic field (if time permits).
Homework:
: Problem on the 3-dimensional harmonic oscillator.

9.

Scattering by a Potential
Reading:
Cohen-Tannoudji, Chapter VIII
Complements A_VIII-C_VIII

Lecture topics:

(a): Scattering by a short-range potential.
(b): Integral equation for scattering.
(c): Scattering amplitude and cross section.
(d): Born approximation.
(e): Scattering by a Yukawa potential.
(f): Scattering phase shifts.
(g): Coulomb scattering.
Homework:
: Complement G_X - Problem 6; others to be determined.

10.

Time-Independent (Stationary State) Perturbation Theory
Reading:
Cohen-Tannoudji, Chapters XI-XII; Feynman, Chapter 12
Complements A_XI, B_XI, E_XI; A_XII-C_XII

Lecture topics:

(a): Review of the exactly-soluble two-state system.
(b): Basic assumptions of time-independent perturbation theory.
(c): Perturbation of a nondegenerate level; first and second order results.
(d): Degenerate perturbation theory; comparison with two-state example.
(e): Fine and hyperfine structure of the H-atom.
(f): Variational method.
(g): Comparison of perturbation methods for the ground state energy of the He-atom.
Homework:
: Complement H_XI - Problems 1,3,5,9.

11.

Time-Dependent Perturbation Theory; Interaction of Light with Matter
Reading:
Cohen-Tannoudji, Chapter XIII; K. Lane notes
Complements D_X, E_X; A_XIII--D_XIII

Lecture topics:

(a)

General solution for H(t) = H₀ + H₁(t); interaction picture; evolution operator U_I(t,t₀).

(b)

Harmonic perturbation; transition rate for $H_1(\vec r, t) = V(\vec r) \sin{\omega t}$ .

(c)

First-order transitions:

i.: Fermi's golden rule; density of states (phase space).
ii.: Differential cross section (Born approximation again).
iii.: Decay rates.

(d)

Two approaches to describing EM interactions of atoms, etc.:

Semiclassical: Quantize radiating system (atom, e.g.), but treat EM radiation field classically.
Fully quantized: Quantize both the radiating system and the EM field (photons).

For emission and absorption amplitudes to O(e), both procedures give the same result. We start with the classical description of radiation for absorption and induced emission, then give the quantized-EM approach to discuss spontaneous emission.

(e)

Classical description of the radiation field $\vec A(\vec r, t)$ .

(f)

Interaction Hamiltonian for a spinless particle of mass m and charge e:

i.: Gauge-invariant momentum--covariant derivative.
ii.: The EM current; conservation law.

(g)

Absorption of light:

i.: Transition amplitude and rate by Fermi's golden rule.
ii.: Detailed balance: $\Gamma({\rm absorption}) = \Gamma({\rm induced \;\; emission})$ .
iii.: Absorption cross section.

(h)

Quantized EM radiation field:

i.: Photon states; creation and annihilation operators.
ii.: Interaction Hamiltonian.
iii.: Spontaneous emission (decay) rate.

(i)

Einstein's A and B coefficients.

(j)

Details of spontaneous emission:

i.: Comparison of quantum and classical results for EM power emitted by an oscillating current.
ii.: Conservation of momentum in EM transitions.

(k)

Electric dipole transitions:

i.: Formalism.
ii.: Selection rules.
iii.: Dipole sum rule.
iv.: Examples from the H-atom and charmonium.

(l)

Scattering of light (nonrelativistic Compton scattering on a spin-0 target):

i.: Thomson cross section.
ii.: Raman scattering.

Homework:

Complement E_XIII - Problems 1,3,5; possibly others.

12.

Identical Particles
Reading:
Cohen-Tannoudji, Chapter XIV
Complement A_XIV

Lecture topics:

(a): Identical particles in classical and quantum mechanics.
(b): Symmetrization postulate for bosons and fermions.
(c): Pauli exclusion principle.
(d): Scattering of identical particles.
Homework
: Complement D_XIV - Problems 4,5,6.

13.

Introduction to Relativistic Quantum Mechanics - Dirac Equation and Solutions.
Reading:
K. Lane notes; Bjorken and Drell, Vol. 1, Chapters 1-3.

Lecture topics:

(a): ``Derivation'' of the Dirac equation for a free particle of mass m; Dirac matrices and spinors.
(b): Dirac equation in covariant form; Dirac gamma matrices; Lorentz transformation properties of the Dirac spinor wave function.
(c): The conserved (electromagnetic) current.
(d): Free-particle solutions (spinors) to the Dirac eqaution; helicity spinors.
(e): Dirac particle in an external EM field; nonrelativistic limit; Pauli equation; $g = 2 + O(\alpha)$ .
(f): Antiparticles and charge conjugation.
Homework:
: To be assigned.

Kane Lane
9/1/1999