/FYS4480

FYS4480-Quantum mechanics for many-particle systems

Quantum Mechanics for Many-Particle systems

This course gives an introduction to the quantum mechanics of many-body systems and the methods relevant for many-body problems in such diverse areas as atomic, molecular, solid-state and nuclear physics, chemistry and materials science. A theoretical understanding of the behavior of quantum-mechanical many-body systems, that is, systems containing many interacting particles - is a considerable challenge in that, normally, no exact solution can be found. Instead, reliable methods are needed for approximate but accurate simulations of such systems.

The aim of this course is to present some of the most widely used many-body methods, starting with the underlying formalism of second quantization. The topics covered include second quantization with creation and annivilation operators, Wick's theorem, Feynman diagram rules, microscopic mean-field theories (Hartree-Fock and Kohn-Sham theories), many-body perturbation theory, large-scale diagonalization methods, coupled cluster theory, algorithms from quantum computing, and Green's function approaches. Both fermionic and bosonic systems are discussed, depending on the interests of the participants. Selected physical systems from various fields such as quantum chemistry, solid-state physics and nuclear physics are studied, depending on the background and interests of the participants.

Instructor information

  • Name: Morten Hjorth-Jensen
  • Email: morten.hjorth-jensen@fys.uio.no
  • Phone: +47-48257387
  • Office: Department of Physics, University of Oslo, Eastern wing, room FØ470

Practicalities

  1. Four lectures per week, Fall semester, 10 ECTS. Thursday 1015am-12pm and Friday 1015-12pm. The lectures will be recorded and linked to this site;
  2. Two hours of exercise sessions for work on projects and exercises, Friday 1215pm-2pm;
  3. Two projects which are graded and count 30% each of the final grade and a final oral exam that counts 40% of the final grade;
  4. Weekly assignments which are not graded;
  5. The course is offered as a so-called cloned course, FYS4480 at the Master of Science level and FYS9480 as a PhD course;
  6. Weekly email with summary of activities will be mailed to all participants;

Textbooks

Recommended textbooks: In the folder https://github.com/ManyBodyPhysics/FYS4480/tree/master/doc/Literature you will find the textbooks we will be following. Weekly reading assignments based on these texts will be sent before each week. In particular we recommend the texts by

Topics (not all will be discussed)

  • Intro chapter with basic definitions and simple examples and mathematics of many-body functions
    • Definitions of SDs etc, permutation operators,linear algebra reminder including reminder about determinants, vector and mtx algebra, tensor products, representations, unitary transformations, link to quantities like
    • one-body and two-body densities, rms radii etc. Discuss ansatze for wave functions and more.
    • Ansaztes for wave functions
  • 2nd quantization for bosons and fermions and more
    • Commutation rules and definition of creation and annihilation operators
    • Proof of wick's theorem
    • Discuss Wick's generalized theorem
    • particle-hole picture
    • interaction, Schroedinger and Heisenberg pictures, pros and cons
    • time dependent wick's theorem
    • Gell-Man and Low's theorem
    • Adiabatic switching
    • Derivation of expressions for different parts of Hamiltonians, 1b, 2b, 3b etc
    • Wigner-Jordan transformation and 2nd quantization
    • Baker-Campbell-Hausdorf (BCH)
    • Suzuki-Trotter as an approximation to BCH
  • FCI, diagrams and particle-hole representations
    • Basics of FCI
    • Rewriting in terms of a particle-hole picture
    • Slater determinants and similarity transformations and algorithms for solving eigenvalue problems
    • Eigenvector continuation
    • Introduce a diagrammatic representation
  • Mean-field theories
    • Hartree-Fock in coordinate space and 2nd quantization
    • Thouless theorem
    • Slater dets in HF theory
    • Density functional theory
    • The electron gas as example
    • FCI and HF, diagrammatic representations and critical discussions
  • Many-body perturbation theory
    • Time dependent and time-independent representations
    • Brillouin-Wigner and Rayleigh-Schrødinger pert theory
    • Diagrammatic representation
    • Linked-diagram theorem based on time-dependent theory
  • Coupled cluster theories, standard and unitary
    • Derivation of equations for singles and doubles, reminder on unitary transformations
    • Unitary coupled cluster theory
  • Green's function theory and parquet theory
  • SRG and IMSRG
  • Monte Carlo methods
    • Taught in FYS4411
  • Quantum computing
    • VQE and unitary CC
  • Time-dependent many-body theory
  • Applications to different systems like the electron gass, Lipkin model, Pairing model, infinite nuclear matter, and more

Teaching schedule with links to material

Week 34, August 21-25

Week 35, August 28-September 1

Week 36, September 4-8

Week 37, September 11-15

Week 38, September 18-22

Week 39, September 25-29

Week 40, October 2-6

Week 41, October 9-13, 2023

Week 42, October 16-20, 2023

Week 43, October 23-27, 2023

Week 44, October 30-November 2, 2023

Week 45, November 6-10, 2023

Week 46, November 13-17, 2023

Week 47, November 20-24, 2023

Week 48, November 27-December 1, 2023

Week 50, December 19, 2023

  • Final oral exam