AA 2021-22, Michele Correggi

The main part of the course is devoted to a synthetic presentation of the postulates of quantum mechanics and its realization as operator theory in Hilbert spaces. More precisely, we first discuss the notions of self-adjointness, spectrum and spectral measure of a symmetric linear operator and then state the spectral theorem, in order to represent any observable as a multiplication operator. Quantum dynamics is then established via the Stone theorem and the self-adjointness of the Hamilton operator. The abstract properties mentioned above are then concretely verified in physical examples, as the free particle and the hydrogen atom. The second part of the course is then focused on some more specific questions about quantum mechanics and, in particular, the classical limit and the behavior of many-body systems of identical particles. The derivation of the classical behavior for a quantum system in the limit of vanishing Plank’s constant is indeed a very relevant question is several areas of quantum physics and it has given rise to a very rich mathematical research line. Similarly, the effects of particle statistics (Bose- or Fermi-Dirac) on many-body quantum systems in suitable mean-field or semiclassical regimes is a flourishing line of research in modern mathematical physics. We plan to give some insights on the more recent advances on these topics, as well as an introduction to more exotic phenomena related to two-dimensional models (anyons).