Uwe Thumm Spring/Fall 2011

Quantum Mechanics 1 – Topics covered in class

1. Introduction

a. Experiments that reveal the particle character of light (Photo-, Compton-effect, black-body radiation)

b. Experiments that reveal the non-classical character of (quasi) particles (Franck-Hertz, H spectrum, Josephson effect).

c. Diffraction of particle beams (single-, double-slit, coherence).

2. Schrödinger Equation for Free Particles

a. Wave function (time independent) and Fourier transforms.

b. Time-dependent wave functions, (Gaussian) wave packets.

3. Schrödinger Equation for a Particle in a Conservative Force Field

a. Consistency with 2a) for V(x, t) = const.

b. Relation to classical mechanics and geometrical optics (Maxwell, Hamilton-Jakobi, eikonal equations).

c. Continuity equation.

d. Schrödinger equation in momentum space.

4. Linear Operators and Expectation Values

a. Definitions (linear, Hermitian operatiors, scalar product, algebra, Hilbert space, square integrability).

b. Momentum operator in coordinate representation.

c. Classical and quantum observables, their (not unique) correspondence and time evolution. Ehrenfest theorem, uncertainty relation. Virial theorem. Time inversion

d. Stationary states. Degeneracy.

e. Expansions in terms of stationary states. Analytic functions of operators. Completeness.

f. Normalization of continuum wave functions.

g. Unitary operators (examples: displacement, time-evolution,...)

h. Charged particles in electro-magnetic fields.

i (active and passive) Galilei transformations.

5. Linear Harmonic Oscillator

a. Stationary states and energies.

b. Time-dependent solutions. Expectation values and comparison with classical solution (in phase space).

6. Simple Model Potentials (1 D)

a. Quantum phenomena at step potentials (transmission, reflection, penetration).

b. Quantitative solutions (step, well, barrier).

c. Scattering: M and S matrices in relation to incident, transmitted, reflected flux. Resonances.

7. Spherically Symmetric Potentials

a. Orbital angular momentum operator. Commutation relations. Relation to rotations. Ladder operators.

b. Eigenvalues and vectors of and L_z. Properties of (associated) Legendre polynomials and spherical harmonics.

c. Radial Schrödinger equation for a free particle. (Spherical) Bessel and Neumann functions.

d. Examples: Coulomb potential. Hydrogen: spectrum and eigenfunctions. Expectation values. Properties of Laguerre polynomials. Momentum-space representation.

8. Perturbation Theory with Applications

a. Time-independent (degenerate) perturbation theory.

b. Valence spectra of alkali atoms.

c. He atom.

d. Stark effect (linear and quadratic) and polarizability for H. Metastable quenching in weak el. field.

e. Time-dependent perturbation theory. Transition rates. Continuum transitions. Fermi’s golden rule.

f. Absorption and induced emission of electro -magnetic radiation. Detailed balancing. Application to hydrogen atoms: Dipole selection rules. Cross sections.

g. Photoelectric effect.

9. Scattering at a Central Potential

a. Scattering amplitude and (angle-differential) cross section. Classical versus quantum interpretation.

b. Green’s functions (inclusion of boundary conditions, contour integration technique).

c. Born series. First Born approximation cross section for Yukawa and Coulomb potentials.

d. Partial wave expansion of scattering wavefunction, amplitude, and cross section.

e. Scattering phase shifts and resonances. Example: scattering off radial square-well potential.

Uwe Thumm Fall 2011

Quantum Mechanics 2 – Topics covered in class

1. Formal Foundation of QM

a. Hilbert space. Definition, examples. Dirac (bra-ket) notation

b. Linear operators and their representation. Normal, Hermitean, unitary, adjoint operators

c. Operators with continuous spectra. Completeness, closure relation. Functions of operators. Fourier transformation Examples

d. Solving eigenvalue problems by i) matrix diagonalization and ii) variation. Example: He atom

e. Observables and measurement. Single & simultaneous measurements, complete sets of commuting observables.

2. Application of Algebraic Operator Techniques

a. Harmonic oscillator. Quantization of classical fields. Quasi particles

b. Coherent and squeezed states

c. Angular momentum algebra

3. Quantum Dynamics

a. Time-evolution operator

b. Schrödinger,- Heisenberg-, and Interaction Picture

c. Correspondence principle, quantization

d. Canonical quantization

e. Forced harmonic oscillator: advanced & retarded Green’s functions, time ordering, relation to scattering theory

f. Coordinate representation of the time-evolution operator. Examples: free particle, harmonic oscillator

g. Path integrals. Classical limit. Semi-classical approximations

h. Statistical mixtures of quantum states. Density operator. Shannon and von Neumann entropy

4. Spin

a. Experimental evidence. Stern-Gerlach, Einstein-de Haas experiments. Zeeman effect

b. Math. description: Two component spinors. Remarks on Dirac equation

c. Spin rotations: Pauli matrices

d. Spin dynamics. Spin-orbit coupling. Paramagnetic resonances. Remarks on NMR

e. Spin-dependent scattering: angle-differential cross section, spin-polarization of scattered particles

f. Information content in spin ensembles: von. Neumann and outcome entropy

g. Measurement and reduction of quantum states. Multiple Stern-Gerlach experiments

5. Addition of Angular Momenta

a. Two spin-1/2 particles: product and total spin basis; hyperfine interaction in H (21cm line).

b. Addition of two arbitrary angular momenta: Clebsch-Gordon coefficients, spectroscopic notation

c. (Irreducible) representations of rotations and tensor operators. Wigner-Eckart theorem

d. Applications of Wigner-Eckart theorem: normal & anomalous Zeeman effect, Paschen-Back effect

e. Other symmetry operations: parity, time reversal, translation, iso-spin (example: nuclear spectra).

6. Many-Particle Systems

a. Identical particles, Fock space, symmetrization postulate, bosons, fermions, spin-statistics theorem

b. System of N independent indistinguishable particles (bosons or fermions)

c. System of N interacting indistinguishable particles (bosons or fermions), “second quantization”

d. Direct and exchange interactions, examples: energies, He atom, scattering

e. Hartree-Fock method, Brillouin and Koopman theorem

f. Molecular bonds (example: H₂⁽⁺⁾)

7. Relativistic Quantum Mechanics

a. Klein-Gordon equation for spinless particles

b. Dirac equation for spin-½ particles

c. Electromagnetic interactions of Dirac particles. Pauli equation for small <v>/c. Gyromagnetic factors

d. Approximate solution of Dirac equation for H atom

e. Outline of exact solution of Dirac equation for H atom and QED corrections