Contents

Quantum mechanics for many-particle systems

Quantum mechanics for many-particle systems

Introduction

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.

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 and diagrams and particle-hole representations

    • Basics of FCI

    • Rewriting in terms of a particle-hole picture

    • Discuss 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

    • DFT links

    • The electron gas as example

    • FCI and HF, diagrammatic representations and critical discussions

  • Many-body perturbation theory

    • Time dependent and time-independent representation

    • 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

    • non-hermiticity

    • Specialize to CCD case and compare with FCI and MBPT

  • 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

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Schedule