Department of Physics and Astronomy

Syllabus: Physics 8164

1. Introduction

Examples for non-relativistic quantum many-particle systems: atoms, molecules, condensed matter problem, atomic nuclei, stellar astrophysics. Review: total energy (Hamilton function) for N particles in classical mechanics, and quantization rules. Structure of Hamiltonian for quantum many-particle systems: the N-electron atom as example. One- and two-body operators. Time-dependent Schroedinger equation for quantum many-particle systems; need for approximate solution. The Hamiltonian for condensed matter systems, and for nuclear structure. Remarks about astrophysics problem (interior of a star).

2. Quantum many-particle systems in coordinate space and in occupation number space

Systems of identical particles. Symmetric and anti-symmetric many-particle wave functions. Fermions, bosons, and Pauli's spin-statistic theorem. Non-interacting systems (one-body Hamiltonians): examples of free Fermi gas, N-electron atom (neglecting e- e- interaction), nuclear shell model. Total energy and many-body wavefunction for one-body Hamiltonians. Transition from coordinate space to occupation number space, creation and annihilation operators for fermions (anti-commutation relations). Quantum field operators. One-body, two-body operators, and three-body operators in occupation number space ("second quantization"). Examples for one-body operators: particle density operator for fermions, operators for total linear momentum and angular momentum. Formal expression for exact ground state expectation values of these operators.

3. Non-interacting ground state and particle-hole transformation

Non-interacting ground state for one-body Hamiltonians. Canonical particle-hole (p-h) transformation (= simplest quasi-particle transformation). The non-interacting ground state as quasi-particle vacuum. Excited states: 1p-1h, 2p-2h, etc. Expectation values of one-body operators in non-interacting ground state: density for free Fermi gas and for N-electron atom. The free electron gas in second quantization: Fermi momentum, ground state energy per particle.

4. Quantum many-particle theory and quantum field theory, Wick's theorem

Equivalence of quantum many-particle theory to (non-relativistic) quantum field theory. Time- and normal-ordered products of field operators, contractions of field operators. Wick's theorem for time-dependent field operators. Special case: Wick's theorem for time-independent creation / annihilation operators; use to calculate expectation value of 2-body operator in non-interacting ground state.

5. The interacting electron gas

Interacting electron gas (model for metal or plasma): Hamiltonian in first and second quantization. Compute ground state energy per particle in first-order perturbation theory using Wick's theorem. Direct and exchange energies. Comparison to exp. data (metallic sodium); low-density limit: Wigner solid.

6. Density matrix and pair correlation function for fermions and bosons

Definition of density matrix and pair correlation function in terms of field operators. Evaluation of pair correlation function using Wick's theorem. Pair correlation function for free fermions and bosons.

7. Mean field approximation: static Hartree-Fock (HF) and density functional theory

Basic idea of mean field approximation: derive 1-body "mean field potential" from given 2-body interaction. Time-dependent Schroedinger equation for N identical fermions in coordinate space and in occupation number space. Static N-body Schroedinger equation and equivalence to Ritz variational principle for energy functional. Main approximation: use single Slater determinant. Brief history of HF theory: Hartree potential for atoms (1928), Fock exchange term (1930). Derivation of static HF equations from Ritz variational principle. HF ground state energy and mean field Hamiltonian. Self-consistent HF equations in energy and coordinate representation. Iterative numerical solution of HF equations. Structure of energy density functional predicted by HF theory.
Numerical results: a) Standard Hartree-Fock calculations of atomic electron densities which reveal the electron shell structure. b) Multi-Configuration Hartree-Fock / Dirac-Fock calculations of atomic binding energies and transition probabilities; c) Standard Hartree-Fock theory of nuclear ground state properties: nuclear binding energies and proton / neutron density distributions.
Introduction to density functional theory in connection with molecular physics / quantum chemistry and condensed matter physics: Hohenberg-Kohn theorem, and Kohn-Sham formalism. Discuss Physics Today (2015) article: "A half century of density functional theory".

8. Time-dependent Hartree-Fock (TDHF) and Time-dependent density functional theory (TDDFT)

Main assumption of TDHF: use single time-dependent Slater determinant. Two important practical applications: a) atom / nucleus in time-dependent external field; b) collision of two nuclei (fusion and deep-inelastic reactions). Straightforward generalization of static Kohn-Sham formalism to time-dependent external fields.

Guest lecture by Prof. Pantelides: Applications of static and time-dependent density functional theory to condensed matter physics and nanomaterials.

9. BCS theory at zero temperature: Cooper pair formation

Experimental information about low-temperature superconductors. Classical sound waves in fluids: Lagrange and Hamilton density, canonical quantization --> phonons. Acoustic phonons, Debey theory of specific heat. Lattice vibrations (phonons) and electron-phonon interaction. Cooper pair model. Thermodynamic potential at T=0, BCS Hamiltonian, canonical transformation to quasi-particles (Bogoliubov transf.), gap equation, normal and superconducting ground state. Brief discussion of pair formation in atomic nuclei.

10. Perturbative (Green's function) formalism at zero temperature

QM representations: Schroedinger picture, Interaction picture, and Heisenberg picture. Time-development operator. Fermion propagator (Green's function) = ground state expectation value of field operators. Gell-Mann and Low theorem for exact many-body ground state. Feynman-Dyson perturbation expansion of fermion propagator, derive Feynman rules. Feynman diagrams in coordinate space.
Calculate observables: a) ground state expectation values of one-body operators (number density, spin density, and kinetic energy density) using fermion propagator; b) time-dependent version of Goldstone's theorem for ground state energy: derive Feynman graphs in first and second order of perturbation theory.
Dyson's integral equations for exact fermion propagator, including "proper self-energy". Hartree-Fock theory in Feynman-Dyson approach: use proper self-energy in first order, but replace some free propagators with exact propagators (HF includes infinite orders in perturbation theory, i.e. it is nonperturbative).
Feynman diagrams in 4-momentum space (useful in particular for uniform and isotropic media).