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The pillar course in Statistical Mechanics aims to provide the tools for a mathematical description of the equilibrium and non equilibrium properties of systems made by a large number of constituents. While at the microscopic scale, those systems are described by fundamental laws (Newton equations in the classical setting, Schrödinger equations in the quantum case), simpler non linear PDEs allow to capture their collective behaviour at the macroscopic scale. Understanding the validity of those effective equations, as well as their properties, is a major challenge in mathematics, and have lead to the developments of several tools spanning from probability to functional analysis.
More information about Part 2 and 3 of the course can be found at the following links:
Part 2: Probability theory for statistical mechanics
Part 3: Some introductory topics in statistical mechanics
Part 1
The course will introduce the mathematical foundations of Quantum Mechanics. Beginning with key paradigmatic one-particle problems, we will then focus on the mathematical analysis of large systems of interacting quantum particles.
Timetable
The course consists of lectures by Prof. Serena Cenatiempo and Dr. Daniele Ferretti, with three lectures scheduled each week as follows:
Course description
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.
Course requirements
Basic elements of the theory of linear operators in Hilbert spaces.
Course content
A tentative schedule of the course is detailed below.
Nov. 4: An introduction to Quantum Theory [SC]
Nov. 5: Self-adjoint operators. Free particle in a box. [DF]
Nov. 7: One particle models: Harmonic Oscillator [SC]
Nov. 11: Spectrum and Spectral Theorem for self-adjoint operators [DF]
Nov. 12: One particle models: Free particle [SC]
Nov. 14: Symmetries and Spin [DF]
Nov. 19: Evolution of wave packets. Pure and Mixed States [SC]
Nov. 20: Hydrogen Atom. Entanglement. [SC]
Nov. 21: Many particle systems and Second Quantization [DF]
Nov. 25: Reduced Density Matrices [SC]
Nov. 26: Statistical Ensembles in Quantum Mechanics [SC]
Nov. 28: Non interacting Bose and Fermi gases [SC]
Optional classes on Bose-Einstein condensation in interacting Bose gases and the rigorous derivation of the Gross-Pitaevskii equation will be offered in February 2025.
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Lecture Notes
References