PHYS 625, Non-relativistic Quantum Mechanics, Spring 2015
Prof. Victor M. Yakovenko

Textbook: Radi A. Jishi
"Feynman Diagram Techniques in Condensed Matter Physics"
Cambridge University Press, 2014, ISBN 9781107655331

Timeline of the course

Mondays, Wednesdays, and Fridays, 9 - 9:50 am, room 1219 in Physics Building

The text above and at the update point is the current timeline of the course. The text below the update point is a tentative schedule of the course. The update point will be moving down during the semester, and the timeline will be progressively updated.

Week 1
Monday, January 26 Introductory lecture; Read Ch. 1 (Quantum Mechanics) and Ch. 2 (Single-Particle States in Solids) on your own as a refresher.
Wednesday, January 28 Ch. 3 Second Quantization
3.1 N-particle wave function; 3.2 Fermi and Bose exchange symmetry; Properly symmetrized basis set; Occupations numbers;
Fermions: Slater determinant; 3.4.1 Creation and annihilation operators
Friday, January 30 Matrix elements of one-body (3.5) and two-body (3.7) operators for fermions (A.1)
Week 2
Monday, February 2 Bosons: 3.4.2 Creation and annihilation operators; A.2 Matrix elements of one-body and two-body operators;
HW 1 issued (Second Quantization)
Wednesday, February 4 3.11 Field operators Psi(r); Second quantization in coordinate and momentum representations
Friday, February 6 Discussion of HW 1: Tight-binding models on 1D, square, and hexagonal lattices, graphene;
Spin Hamiltonian for a ferromagnet, the Holstein-Primakoff transformation, and spin waves (magnons)
Week 3
Monday, February 9 3.8, 3.9 Coulomb interaction between electrons in momentum representation
Ch. 4 The Electron Gas
The direct and exchange contributions to the electron gas energy
HW 2 issued (The Electron Gas)
Wednesday, February 11 4.1 Cancellation of the direct Coulomb energy with positive background in the jellium model
4.3 Negative exchange contribution to the Coulomb energy
Friday, February 13 Problem 4.3: Real-space avoidance by fermions as the origin of the exchange contribution;
Variational minimization of the total energy; 4.2 The rs parameter; High- and low-density limits; Wigner crystal;
Problems 4.4 and 4.5: The two-dimensional electron gas (2DEG)
Week 4
Monday, February 16 Ch. 5 A Brief Review of Statistical Mechanics
5.3.3 The grand canonical ensemble; 5.4 The statistical operator and density matrix; 5.5 Fermi and Bose distribution functions
HW 3 issued (Statistical Mechanics)
Wednesday, February 18 Reduced one-particle and two-particle density matrices (Problems 4.2 and 4.3)
Ch. 6 Real-Time Green's and Correlation Functions
6.1 A plethora of functions
Friday, February 20 6.6 Linear response theory; Kubo formula; Retarded correlation function
Week 5
Monday, February 23 6.5 Green's function of a noninteracting system; Analytical properties in the complex plane of omega
HW 4 issued (Real-Time Correlation Functions)
Wednesday, February 25 6.4 Spectral representation; 6.4.3 Retarded correlation function; 6.4.4 Correlation function;
Real and imaginary parts of the correlations functions
Friday, February 27 6.4.4 Fluctuation-dissipation theorem; Problem 6.11 Kramers-Kronig relations
Week 6
Monday, March 2 No lecture: Late campus opening because of ice
HW 5 issued (Real-Time Correlation Functions)
Wednesday, March 4 6.7 Density-density correlation function, the Lindhard function
Friday, March 6 No lecture: Campus closed because of snow
Week 7
Monday, March 9 Problem 2 of HW 5: continuum of electron-hole excitations in 3D, 2D, and 1D; Neutron scattering;
6.9 Paramagnetic susceptibility of a noninteracting electron gas
HW 6 issued (Applications of Real-Time Green's Functions)
Wednesday, March 11 6.10 Equation of motion; The hierarchical chain of correlation functions; 6.12 A localized state coupled to a continuum of states
Friday, March 13 Ch. 7 Applications of Real-Time Green's Functions
7.1 Green's function for a single-atom Hubbard model; upper and lower Hubbard bands; 7.2 Anderson's impurity model;
Mean-field approximation; Models of magnetism for one impurity, localized spins 1/2, and itinerant Hubbard/Stoner model
(Problems 2, 3, and 4 in HW 6). Do Problem 1 about tunneling (7.3) on your own.
Week 8
Monday, March 23 Ch. 8 Imaginary-Time Green's and Correlations Functions
8.1, 8.2, 8.3 Periodicity in Matsubara time, even and odd Matsubara frequencies;
8.4 Spectral representation, analytical continuation, and relation to real-time functions
HW 7 issued (Imaginary-Time Green's Functions)
Wednesday, March 25 8.5 Green's function for noninteracting particles
8.7.1 The interaction picture
Friday, March 27 8.7.2 The U-operator; Time-ordered perturbation expansion; 8.7.3 Green's function in terms of the U-operator;
8.7.4 Perturbation expansion of the imaginary-time Green's function
Week 9
Monday, March 30 8.8 Wick's theorem
HW 8 issued (Diagrammatic Techniques)
Wednesday, April 1 8.9 Case study: first-order interaction; 8.10 Cancellation of disconnected diagrams
Friday, April 3 Ch. 9 Diagrammatic Techniques 9.1 Second-order perturbation;
Feynman rules in momentum-frequency (9.2) and coordinate (9.4) spaces;
Summation over omega_n using contour intergration
HW 9 issued (Electron Gas: A Diagrammatic Approach)
Week 10
Monday, April 6 Factor (-1) for fermion loops; Cancellation of 1/2 and 1/n!;
9.5 Self energy and Dyson's equation; 9.6 Energy shift and the lifetime of excitations
Skip 9.7 Time-ordered diagrams; 9.8 Dzyaloshinski's rules
Wednesday, April 8 No lecture: Victor Yakovenko is away
Friday, April 10 Ch. 10 Electron Gas: A Diagrammatic Approach
10.4 Classification of diagrams according to the degree of divergence
Victor Yakovenko is away, substituted by Prof. Jay Sau
Week 11
Monday, April 13 10.6 Summations of the ring diagrams; 10.7 Screened Coulomb interaction; 10.8 Collective electronic density fluctuations
Victor Yakovenko is away, substituted by Prof. Jay Sau
Wednesday, April 15 10.10 Dielectric function; 10.10.1 Thomas-Fermi model;
10.11 Plasmons and Landau damping; 10.11.1 Plasmons; 10.11.2 Landau damping
Skip 10.12 Dielectric function of graphene
Friday, April 17 Ch. 11 Phonons, Photons, and Electrons
11.1 Lattice vibrations in 1D; 11.2 1D diatomic lattice; acoustic and optical phonons; 11.3 Phonons in 3D; 11.4 Phonon statistics;
Victor Yakovenko is away, substituted by Prof. Jay Sau
Week 12
Monday, April 20 11.5 Electron-phonon interaction: rigid-ion approximation
HW 10 issued (Phonons, Photons, and Electrons)
Wednesday, April 22 11.7 Phonon Green's function; 11.8 Free-phonon Green's function;
11.9 Feynman rules for the electron-phonon interaction; 11.10 Electron self energy
Friday, April 24 Phonon self energy; Peierls instability in 1D; Charge-density waves;
The Su-Schrieffer-Heeger model of polyacetylene (CH)n
Week 13
Monday, April 27 11.11 Quantization of the electromagnetic field; 11.12 Electron-photon interaction; Gauge invariance
Skip 11.13 Light scattering by crystals; 11.14 Raman scattering in insulators
Wednesday, April 29 Ch. 12 Superconductivity
12.1 Properties of superconductors; 12.2 The London equation; the Meissner effect; the London penetration depth
HW 11 issued (Superconductivity)
Friday, May 1 12.3 Effective electron-electron attraction mediated by phonons; Summation of the Cooper ladder diagrams
Week 14
Monday, May 4 Divergence of superconducting susceptibility at the transition temperature Tc (Problem 1 of Homework 11);
Mean-field description of the superconducting state at T<Tc
Wednesday, May 6 The Bogoliubov-de Gennes equations; Diagonalization of the 2x2 Hamiltonian by a unitary transformation to the
Bogoliubov operators; The BCS theory; Energy gap in the spectrum of quasiparticles (Problems 2 and 3 of Homework 11)
Friday, May 8 Energy minimization with respect to Delta; Energy gain at T=0; Delta0 at T=0 and the universal ratio Delta0/Tc;
General equation for Delta(T); The free energy F(Delta,T) and spontaneous symmetry breaking
(Problem 3 of Homework 11)
Week 15
Monday, May 11 12.7 Green's function approach to superconductivity; 12.9 The Nambu formalism;
12.10 Response to a weak (electro)magnetic field
Final Exam
Monday, May 18, 8-10 am
update point
Make-up lecture for the snow days in lieu of final exam
Gauge-invariant expression for supercurrent; vortices in superconductors; magnetic flux quantization;
type II superconductors; mixed state; lower and upper critical magnetic fields;
Ch. 13 Nonequilibrium Green's Functions
13.2 Schroedinger, Heisenberg, and interaction pictures; 13.3 The malady and the remedy; Diagram technique at T=0;
13.4 Contour-ordered Green's function; 13.5 Kadanoff-Baym and Keldysh contours;
13.6 Dyson's equation; 13.7 Langreth rules; 13.8 Keldysh equations

Last updated May 18, 2015