Phys 712 — Spring 2015
Department of Physics & Astronomy
Instructor Dr. Kevin Beach
Office: 206 Lewis Hall

More details provided in the syllabus.


  1. due Friday, January 30; solution to question 3(b)
  2. due Wednesday, February 18
  3. due Friday, March 6; please only submit questions 2 and 3 for grading
  4. due Friday, April 15


  1. Review of the syllabus; differences between classical and quantum mechanics in the Lagrangian and Hamiltonian schemes; features of quantum mechanics that break our classical intuition; Dirac notation; linear vector spaces; orthonormality and completeness of states
  2. Operator formalism; Hilbert space axioms; operators and outer products; self-adjointness; time evolution of a quantum state; basis of energy eigenstates; unitarity of the evolution operator; Schrödinger and Heisenberg pictures; Dyson series; general case of evolution as a time-ordered exponential
  3. Quick review of 1D wave mechanics; free particles; Fourier transform relation between position and momentum; evolution operator evaluated with Suzuki-Trotter time slices; formal connection to Feynman path integral
  4. Propagators for free particle, harmonic oscillator, and generic single-particle models; derivations via operators and path integrals—see Sakurai Ch. 2.6
  5. Lagrangian for a charged particle in a classical EM field; Aharanov-Bohm effect—see Sakurai Ch. 2.7
  6. Cancelled due to University shutdown
  7. Review of Assignment 1; pole structure of propagators; interaction picture (as distinct from the Schrödinger and Heisenberg pictures)—see Sakurai Ch. 5.5
  8. Interaction picture; time dependence of states and operators; expansion coefficients obey a coupled set of first order differential equations; example: Rabi oscillations—see Sakurai Ch. 5.5
  9. Dyson equation; formulation of time-dependent perturbation theory in terms of the evolution operator in the interaction picture; transition probabilities; kicked harmonic oscillator; harmonic perturbations—see Sakurai Ch. 5.7
  10. Scattering of free particles from a time-dependent barrier in 1D; sudden approximation; adiabatic approximation—see Sakurai Ch. 5.6
  11. Importance of gapped versus gapless excitations and level crossings for adiabatic evolution; shaken harmonic oscillator; lattice translation; simultaneous eigenstates of definite energy and crystal momentum; tight-binding model of electron hopping around a ring
  12. Dr. Luca Bombelli
  13. Dr. Luca Bombelli
  14. Dr. Luca Bombelli
  15. Cancelled due to University shutdown
  16. Dr. Luca Bombelli
  17. Continuous and discrete translational symmetry; momentum and crystal momentum eigenstates;
  18. Cancelled
  19. Discrete translational symmetry; diagonalization of the tight-binding model via discrete Fourier Transform; block diagonal structure in the case of a nontrivial basis in each unit cell, e.g. graphene; symmetries in systems of quantum spins
  20. Spin-rotation invariance; spin sectors; addition of spin angular momentum; block structure of the spin Hamiltonian; spin-flip and time-reversal invariance
  21. Review of symmetry operators, good quantum numbers, selection rules, and block diagonalization; time-reversal operator; anti-unitarity; time reversal of free, spin-half particles
  22. Time-reversal symmetry and Kramer’s degeneracy; hyperfine splitting of Hydrogen and Deuterium; time reversal in the language of path integrals and propagators
  23. Many-body quantum states; expanded Hilbert space for a tight-binding model with two or more electrons; indistinguishability; particle exchange symmetry for fermions and bosons; Helium atom
  24. Helium atom with perturbative treatment of the e–e Coulomb repulsion; excited states; stabilization of para- and orthohelium; effective spin interactions arising from purely Coulombin forces; fermionic creation and annihilation operators
  25. Second quantization; number operator formalism; states of definite occupation; bilinear Hamiltonians; many-bond wave functions built as products of single-particle wave functions; contractions and Slater Determinants
  26. Second quantized form of the Hamiltonian for fermions and boson; pair-wise interaction terms
  27. Generalization of tight-binding electrons to higher spatial dimension, inifinite lattice sites, and continuum propagation; ground state of a gas of noninteracting electrons; fermi sea, fermi wave vector, and fermi energy