Physics 821: Advanced Quantum Mechanics


Spring semester 2010


Instructor:  Prof. Dmitri V. Khveshchenko

296 Phillips Hall, 962-7213,      

Lectures:       TTh 1:30-2:45 PM, 277  Phillips 

Office hours:  before and after class, 296 Phillips


In many areas of Physics that include High Energy, Nuclear, and Condensed Matter, the understanding of the essential
physical phenomena requires a consideration of the collective effects of a large number of degrees of freedom.
Quantum Many-Body Theory is the tool as well as the language that has been developed to
describe the physics of problems in such apparently dissimilar fields. This course is designed to provide the basic conceptual
and computational tools of Quantum Many-Body Theory to students (both theorists and experimentalists)
with a wide range of interests in Physics. This material will also be used in subsequent and more specialized
courses (e.g., Phys822/823). As a prerequisite, all the students should take Phys721/722 (Quantum Mechanics).

Grading will be based on the homework and students' participation in the discussions of the course material.


Excerpts from the AGD textbook are cited as, e.g., "AGD_1.10-1.12"
which stands for Chapter 1, parts 10 to 12.

1. Non-relativistic boson/fermion many-body systems:

  1.1  Creation/annihilation operators, Fock space, ground and excited states of interacting bosons/fermions  (AGD_1.3-1.5)
  1.2  Interaction representation, Green function, S-matrix. (AGD_2.6, 2.7)
1.3  Perturbation theory, Wick theorem, Feynman diagrams, Dyson equations (AGD_2.8-2.10)
  1.4  Self-energy and polarization operator, Example of Coulomb interacting fermions (AGD_4.22)
  1.5  Thermal Green function, Matsubara representation, Thermodynamic potential.
   2. Introduction to relativistic field theory:

   2.1 Classical equations of motion, Lagrangians and actions, canonical quantization.(F_Ch.2.3-2.5, 4.2, 4.4-4.5)
   2.2 Discrete/continuous and global/local symmetries, conservation laws, Noether's theorem.  (F_Ch.3.1-3.4)
Free scalar and vector fields, Klein-Gordon  and Maxwell equations. (F_1.1-1.2)
   2.3 Dirac equation, (bi)spinors and Grassmann algebra. (F_Ch.2.5-2.7, 7.1)

   3. Path integral techniques:     

   3.1 Path integral in single-particle quantum mechanics. Harmonic oscillator. (F_Ch.5.1-5.2)
   3.2 Path integral for a relativistic bosonic field.
 . (F_Ch.5.3-5.6)

   3.3 Gaussian path integrals. Perturbation theory and cluster expansion.

   3.4 Grassmanian numbers. Fermion path integral

  4. Further applications of many-body techniques:

  4.1 Interacting non-relativistic fermions: Hartree-Fock and Random Phase Approximations.

  4.2 Interacting non-relativistic bosons: Bogoliubov transformation, superfluidity and superconductovity.

  4.3 Landau theory of phase transition: mean field and fluctuations. Poor man's scaling.
  4.4.Fermion and spin chains, Jordan-Wigner transformation and bosonization, solitons and kinks.
  4.5 Quantum statistical mechanics, Matsubara sums, thermal Green functions (AGD_3.11-3.16) .
  4.6 Response and correlations functions, fluctuation-dissipation theorem, quantum critical behavior.


Concise introduction to the subject:

(AGD) A. Abrikosov, L. Gorkov and I. Dzyaloshinski. ``Methods of Quantum Field Theory in Statistical Physics", Dover.

Online resources:

(F) General field theory course by E.Fradkin (UIUC):

(B) A similar course - with emphasis on particle physics - by I.Balitsky (ODU):

Other useful textbooks:

M. Stone, "The Physics of Quantum Fields" , Springer.
P.Ramond, "Field Theory : a Modern Primer", Addison Wesley

G. Mahan, "Many Particle Physics", Plenum Press.
A. Fetter and J. D. Walecka. ``Quantum Theory of Many Particle Systems", McGraw-Hill.

-More up to date:
E. Fradkin. ``Field Theories of Condensed Matter Systems", Addison-Wesley.
S.Sachdev, "Quantum Phase transitions", Cambridge Univ. Press.
A. M. Tsvelik, "Quantum field theory in condensed matter physics", Cambridge Univ. Press.

Homework assignments
:  see the Phys821_HW file.