Variational principles. Eulero-Lagrange equations. Variational formulation of mechanics., Liouville and Poincare' theorems. Hamiltonian and equivalence Ideal constraints Virtual work and Lagrangian formalism Poisson parenthesis Canonical transformations Hamilton jacobi equation Part 2: Quantum mechanics: Group and phase speed. Crisis of classical mechanics and beginning of quantum mechanics Introduction of Schroedinger equation Operators in Hilbert spaces: bounded operators, adjoint of an operator, selfadjoint operators, weak derivative. Eigenfuncions of the Schroedinger operator. Regularity of the eigenfunctions of the Schroedinger operator Examples: free particle, potential well, harmonic oscillator, idrogen atom. Axioms of quantum mechanics. Integrals of motion. Indetermination principle.
Part 3: Statistical mechanics Classical Statistical Mechanics and Statistical ensembles: grand canonical, canonical and microcanonical Thermodynamic quantities in perfect gases and equivalence Specific heat for a gas of diatomic molecules Specific heat of solids and equipartition Van-Hove theorem on the existence of thermodynamic limit Tonks Gas Spin systems and phase transitions Exact solution of the Ising model in one dimension Exact solution of the Ising model with infinite range Quantusm statistical mechanics: bosons and fermions Fermi-Dirac statistics, Fermi surface, specific heat Bose-Einstein statistics Bose Condensation Black body radiation and specific heat of phonons
Prerequisites for admission
Basic knowledge of mathematics and physics
The lectures are traditional and the frequence is suggested.
V. I. Arnold: Metodi matematici meccanica classica.
L.E. Picasso, Lezioni sui Fondamenti della Meccanica Quantistica
D.C. Thompson: Mathematical Statistical Mechanics
Note on the websites of the professors
Assessment methods and Criteria
The final examination consists in a written exam and an oral exam. During the written exam, the student must solve some exercises, with the aim of assessing the student's ability to solve problems. In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve problems.