Prof. Thumm Spring 2004

Quantum Mechanics 1 – Topics covered in class

1. Introduction

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

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

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

d. Stationary states. Degeneracy.

e. Time inversion.

f. Virial theorem.

g. Expansions in terms of stationary states. Analytic functions of operators. Completeness of a basis.

h. Normalization on continuum wave functions.

i. Unitary operators (importance in quantum, examples: displacement, time-evolution,...)

j. Charged particle in electro-magnetic field.

k. (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.

6. Simple Model Potentials (1 D)

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

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

c. Scattering: M or 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. Spherically symmetrical potentials, 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.