This course gives an introduction to the quantum mechanics of many-body systems and the computational 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 no exact solution can be found; instead, reliable methods are needed for approximate but accurate simulations of such systems on modern computers. Besides the intrinsic theoretical interest in such methods, they are of great pratical importance in modern research and industry, in fields such as semi-conductor physics, materials science and many other.

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 and with emphasis on non-relativistic theory. The topics covered are the 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, quantum machine learning and Green's function approaches. Both fermionic and bosonic systems are discussed. Selected physical systems from various fields such as chemistry, solid-state physics and nuclear physics are studied, depending on the background and interests of the participants. Quantum computing algorithms may also be discussed. This course includes also some numerical projects.