Quantum statistical mechanics deals with the study of quantum systems made up by a large number of particles. In particular it aims to understand the macroscopic behaviour of these systems, starting from their microscopic fundamental description. While in the absence of interaction, the properties of many body systems can be deduced from the single particle Hamilton operator, in presence of interactions one needs to study the full N particle Schrödinger equation, with N very large and virtually infinite. This requires the development of new mathematical methods and tools. This course aims at providing a mathematical introduction to the theory of Quantum Mechanics and to many particle problems in quantum mechanics, as a preparation to follow research seminars and advanced courses on this topic.
Basic elements of the theory of linear operators in Hilbert spaces.
Week 1 - Historical introduction & postulates of Quantum Theory
Week 2 - One particle models: Free Particle, Harmonic Oscillator, Hydrogen Atom
Week 3 - Quantum Many-Particle Systems, Statistical Ensembles, Ideal Bose and Fermi gases
E.H.Lieb and R.Seiringer. The Stability of Matter in Quantum Mechanics. Cambridge, 2009.
R. Seiringer. Cold Quantum Gases and Bose-Einstein Condensation. Lecture notes from the school ``Quantum Theory from Small to Large Scales'', August 2-27, 2010 [available here]
F. Schwabl. Statistical Mechanics. Springer, 2006.
A. Teta. A Mathematical Primer on Quantum Mechanics. Springer, 2018.