Mathematical Physics 3

A.Y. 2019/2020
Overall hours
Learning objectives
The course aims at providing the basic notions of Hamiltonian, Statistical and Quantum mechanics.
Expected learning outcomes
Undersatnding of fundamentals notions of Hamiltonian, Statistical and Quantum mechanics and ability of solving simple problems on these topics.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
Part 1: Hamiltonian mechanics

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
Axioms of quantum mechanics. Integrals of motion. Indetermination

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
Teaching methods
The lectures are traditional and the frequence is suggested.
Teaching Resources
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.
MAT/07 - MATHEMATICAL PHYSICS - University credits: 9
Practicals: 44 hours
Lessons: 45 hours