Advanced Quantum Mechanics

PHYSICS765

Introduction to Fock space; Fock space approach to non-relativistic quantum mechanics; concepts of quantum fields and canonical quantization; Hamiltonians for relativistic particles; Dirac Hamiltonian and spin-half particles; Hamiltonians for lattice vibrations, phonons, and scalar particles; global symmetries and spontaneous symmetry breaking; local (gauge) symmetry; electromagnetic field quantization and the Hamiltonian for photons; interactions of atoms with photons; density matrix, entanglement and entanglement entropy; interacting bosons and superfluidity; interacting fermions and superconductivity; the path integral formulation; coherent state path integral methods for bosons and fermions. 3 units.

Possible principal texts:

  1. E. Merzbacher, Quantum Mechanics, 3rd ed., (1997).
  2. Greiner and Mueller, Quantum Mechanics: symmetries, (2004).
  3. C. A. Brau, Modern Problems in Classical Electrodynamics, (Oxford Univ. Press, 2004).
  4. J. J. Sakurai, Advanced Quantum Mechanics, (1967).

Other texts to consider:

  1. R. Shankar, Quantum Field Theory and Condensed Matter, (2017).
  2. M. Peskin and D. Schroeder, Quantum Field theory, (1995).
  3. Piers Coleman, Introduction to Many Body Physics (2015).
  4. E. Fradkin, Field Theories of Condensed Matter Physics, 2nd Ed. (2015).

Prerequisites

The prerequisites are Graduate Quantum Mechanics, PHYSICS 764.

Syllabus

  • Fock space formalism
  • Non-relativisitc QM in the Fock space approach
  • Concepts of quantum fields and canonical field quantization.
  • Dirac Hamiltonian in the Fock space and spin-half particles
  • Lattice vibrations and Fock space Hamiltonian for scalar particles
  • Global symmetries: space-time versus internal symmetries, chiral symmetries.
  • Spontaneous symmetry breaking
  • Local (gauge) symmetries and the EM fields
  • Quantization of EM field and photons
  • Interactions of atoms with photons.
  • Density matrix, entanglement and entanglement entropy.
  • Weakly interacting Bose gases, superfluidity
  • Weakly interacting Fermi gases, superconductivity
  • Path integrals approach in the Fock space, bosons and fermions.

Sample lecture schedule

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

  • Lecture 1: Introduction to Fock space formalism
  • Lecture 2: Non-relativistic QM in the Fock space approach
  • Lecture 3: Cancept of a quantum field and canonical quantization
  • Lecture 4: Dirac Hamiltonian in Fock space: set up
  • Lecture 5: Dirac Hamiltonian in Fock space: eigenstates and spin-half particles
  • Lecture 6: Rieview of lattice vibrations  
  • Lecture 7:  Fock space Hamiltonian for vibrations, phonons and scalar particles
  • Lecture 8:  Global Symmetries: Space-time versus internal symmetries
  • Lecture 9: Spontaneous symmetry breaking
  • Lecture 10: Local (gauge) symmetry, simple examples
  • Lecture 11:  Review of classical EM in the Hamiltonian approach
  • Lecture 12: Quantization of the EM field and photons
  • Lecture 13: Gauss Law, physical Hilbert space; Gauge Fixing.
  • Lecture 14: Examples of atom-photon interactions
  • Lecture 15: Density matrix, entanglement and entanglement entropy.
  • Lecture 16: Weakly interacting Bose gases, formalism.
  • Lecture 17: Normal versus superfluid ground states.
  • Lecture 18: Weakly interacting Fermi gases, formalism
  • Lecture 19: Fermi liquid versus superconducting ground states.
  • Lecture 20: Role of frustration, spin-liquids, non-Fermi liquids.
  • Lecture 21: Review of Path integral in QM.
  • Lecture 22: Path integrals for bosons, scalar field theory.
  • Lecture 23: Coherent state path integrals
  • Lecture 24: Path integrals for fermions, Grassmann variables
  • Lecture 25: Special topics.