Quantum Mechanics

Course Number: PHY764

Course description

Angular momentum and symmetries in quantum mechanics from group theory viewpoint; time-independent and time-dependent perturbation theory; path integral formulation; scattering theory; identical particles; applications.

Possible principal texts:

  1. Cohen-Tannoudji, Diu and Laloe, Quantum Mechanics - 2 vols. (Wiley-Interscience)
  2. Shankar, Principles of Quantum Mechanics (Springer)
  3. Greiner, Müller, Quantum Mechanics: Symmetries (Springer)

Other texts to consider:

  1. Sakurai, Modern Quantum Mechanics (Addison-Wesley)
  2. Merzbacher, Quantum Mechanics (Wiley, 3rd edition)

Prerequisites

The prerequisite is at least one semester of a Quantum Mechanics course at the level of R. Shankar's textbook.

In Duke Physics, there is an undergraduate "Quantum Mechanics I" course (PHY 211), with the synopsis: Experimental foundation of quantum mechanics; wave-particle duality; the single-particle Schrodinger equation and the physical meaning of the wave function; methods for studying the single-particle Schrodinger equation; analytical solutions of the harmonic oscillator and hydrogen atom and experimental tests of these solutions; angular momentum and spin systems; and finally the many-particle Schrodinger equation and consequences of identical particles existing in nature.

Syllabus

  • Time-independent perturbation theory.
  • The “real” hydrogen atom.
  • Identical particles, exchange interaction, helium atom.
  • Many-body states, Slater determinant, Hartree-Fock approximation.
  • Variational method: hydrogen molecule, chemical binding.
  • Periodic potential, Bloch waves, band structure.
  • Time-dependent perturbation theory, Fermi’s Golden Rule.
  • Application to two-state system (e.g., spin rotations, NMR).
  • Elementary two-state systems: neutral kaons or neutrino oscillations.
  • Continuous symmetries, Noether’s theorem, rotation group SO(3).
  • Addition of angular momenta, Clebsch-Gordon coefficients.
  • Tensor operators, Wigner-Eckart theorem.
  • SU(2) and its relationship to SO(3), isospin (weak & strong).
  • Path integral formulation of QM: Principles, free particle, semiclassical limit, particle on a circle, Berry’s phase.
  • WKB approximation.
  • Scattering theory: cross section, S-matrix, T-matrix, unitarity, Born approximation, partial waves, optical theorem.

Choice of special topics: quantum information theory, renormalization group, etc.

Sample lecture schedule

(based on 25 lectures each of duration 75 minutes).

  • Lecture 1: Review of undergrad QM, part 1: Hilbert space, states and observables, measurement, uncertainty relation, identical particles, two-state system.
  • Lecture 2: Review of undergrad QM, part 2: Schrödinger equation, harmonic oscillator, angular momentum, “ideal” hydrogen atom.
  • Lecture 3: Time-independent perturbation theory.
  • Lecture 4: The “real” hydrogen atom.
  • Lecture 5: Identical particles, exchange interaction, helium atom.
  • Lecture 6: Many-body states, Slater determinant, Hartree-Fock approximation.
  • Lecture 7: Variational method: hydrogen molecule, chemical binding.
  • Lecture 8: Periodic potential, Bloch waves, band structure.
  • Lecture 9: Time-dependent perturbation theory, Fermi’s Golden Rule.
  • Lecture 10: Application to two-state system (e.g., spin rotations, NMR).
  • Lecture 11: Elementary two-state systems: neutral kaons or neutrino oscillations.
  • Lecture 12: Continuous symmetries, Noether’s theorem, rotation group SO(3).
  • Lecture 13: Addition of angular momenta 1.
  • Lecture 14: Addition of angular momenta, Clebsch-Gordon coefficients.
  • Lecture 15: Tensor operators, Wigner-Eckart theorem.
  • Lecture 16: SU(2) and its relationship to SO(3), isospin (weak & strong).
  • Lecture 17: Path integral formulation of QM: Principles, free particle.
  • Lecture 18: Path integral – semiclassical limit.
  • Lecture 19: WKB approximation.
  • Lecture 20: Path integral – example: particle on a circle, Berry’s phase.
  • Lecture 21: Scattering theory: cross section, S-matrix, T-matrix, unitarity.
  • Lecture 22: Scattering theory: Born approximation.
  • Lecture 23: Scattering theory: Partial waves, optical theorem.
  • Lecture 24: Special topics.
  • Lecture 25: Special topics.