Advanced Quantum Mechanics
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:
- E. Merzbacher, Quantum Mechanics, 3rd ed., (1997).
- Greiner and Mueller, Quantum Mechanics: symmetries, (2004).
- C. A. Brau, Modern Problems in Classical Electrodynamics, (Oxford Univ. Press, 2004).
- J. J. Sakurai, Advanced Quantum Mechanics, (1967).
Other texts to consider:
- R. Shankar, Quantum Field Theory and Condensed Matter, (2017).
- M. Peskin and D. Schroeder, Quantum Field theory, (1995).
- Piers Coleman, Introduction to Many Body Physics (2015).
- E. Fradkin, Field Theories of Condensed Matter Physics, 2nd Ed. (2015).
The prerequisites are Graduate Quantum Mechanics, PHYSICS 764.
- 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.