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.
- 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
- Name: Ruben Guevara
- Email: ruben.guevara@fys.uio.no
- Office: Department of Physics, University of Oslo, Eastern wing, room FØ375
- Four lectures per week, Fall semester, 10 ECTS. Thursday 1015am-12pm and Friday 1015-12pm FØ434 The lectures will be recorded and linked to this site;
- Two hours of exercise sessions for work on projects and exercises, Friday 1215pm-2pm, FØ434;
- 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;
- Weekly assignments which are not graded;
- The course is offered as a so-called cloned course, FYS4480 at the Master of Science level and FYS9480 as a PhD course;
- Weekly email with summary of activities will be mailed to all participants;
Recommended textbooks: There are many good textbooks on many-body physics. In addition to to ourn own collection of lecture notes, we will use material from the following texts
- Dickhoff and Van Neck at https://www.worldscientific.com/worldscibooks/10.1142/5804#t=aboutBook
- Szabo and Ostlund at for example https://www.amazon.com/Modern-Quantum-Chemistry-Introduction-Electronic/dp/0486691861?asin=0486691861&revisionId=&format=4&depth=1
- Shavitt and Bartlett at https://www.cambridge.org/core/books/manybody-methods-in-chemistry-and-physics/D12027E4DAF75CE8214671D842C6B80C
- Carsten Ulrich, Time dependent Density Functional Theory, gives a good link with other many-body methods such as Hartree-Fock theory and time-dependent theories. See https://global.oup.com/academic/product/time-dependent-density-functional-theory-9780199563029?cc=us&lang=en&
- 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