Web-Based Quantum Mechanics Course

• Review: The Fundamental Assumptions of Quantum Mechanics

  • A free particle, one-dimensional wave packets, the Heisenberg uncertainty relations, the time evolution of a free wave packet, the spreading of a wave packet

• Review: Square Potentials

  • A particle in a time-independent scalar potential, potential steps, square wells, d-function potentials

• The WKB Approximation

  • The WKB approximations for bound states

• Mathematical Foundations of Quantum Mechanics

  • Linear vector spaces, Dirac notation, subspaces, linear operators, Hermitian operators, unitary operators, function of operators, representations in state space, change of representation, the eigenvalue problem, commuting observables, the |r> and |p> representations

• The Postulates of Quantum Mechanics

  • Postulates, mean value and root-mean-square deviation, conservation of probability, the evolution operator

• Representations

  • The Schroedinger picture, the Heisenberg picture, the interaction picture, time-dependent perturbation theory, the evolution of the mean value of an observable, interference effects

• The Density Matrix

  • A statistical mixture of states, the density operator, the physical meaning of the density matrix and the density operator

• The Harmonic Oscillator

  • The eigenvalues and eigenfunctions of the 1D harmonic oscillator, the mean value and root mean square deviation of X and P, coherent states, tensor product spaces, the 3D harmonic oscillator

• Two-Level Systems (A Spin 1/2 particle)

  • Spin, a spin 1/2 particle in a uniform magnetic field, a general study of a two-level system

• Two Spin 1/2 particles

  • The tensor product space of two spin 1/2 particles

• Angular Momentum

  • Commutation Relations, basis states, orbital angular momentum, the spherical harmonics

• Symmetries and Constants of Motion

  • The translation operator, the rotation operator, rotation of observables, scalar and vector observables, spinors, the rotation operator for a spin 1/2 particle

Reference: http://electron6.phys.utk.edu/qm1/Modules.htm

• A Particle in a Central Field

  • The asymptotic behavior of the radial functions, two interacting particles.

• Hydrogenic Systems

  • The hydrogen atom, eigenvalues and eigenfunctions, spectroscopic notation, hydrogenic systems.

• Diatomic Molecules

  • Motion of the nuclei, vibrational-rotational levels.

• Addition of Angular Momentum

  • Clebsch-Gordan coefficients, adding more than two angular momenta, rotation matrices, transformation properties of the spherical harmonics, the Wigner-Eckart theorem, the projection theorem.

• Gauge Transformations, Flux Quantization, Propagator

  • Gauge transformations in electromagnetism, the Aharonov-Bohm effect, magnetic monopoles, propagator and path integrals.

• Periodic Systems

  • Periodic and continuous systems, Electrons in a solid.

• Scattering

  • Scattering by a central potential, free spherical waves, partial waves, phase shifts,   the scattering cross section near a resonance, scattering length and effective range, frame transformations, integral scattering equation for stationary states, the Born approximation, the Yukawa potential, the Coulomb potential.

• Approximation Methods for bound states

  • Stationary perturbation theory, the perturbation of degenerate states, the variational method,

• The fine structure and hyperfine structure of H

  • The Dirac equation, the fine structure Hamiltonian, the hyperfine structure Hamiltonian.

• Time Dependent Perturbation Theory

  • Atoms interacting with an electromagnetic wave, transition selection rules

• Identical Particles

  • Permutation operators, Slater Determinants, atomic spectra, coupling schemes