Learning outcomes of the course unit
To provide the basic theoretical concepts in Lagrangian and Hamiltonian mechanics. To understand the principles leading to the study of macroscopic systems and to discuss the basic concepts in statistical mechanics and the methods to calculate the thermodynamical properties of macroscopic systems at equilibrium, starting from the statistical distribution of microscopic variables in phase space.
Acquiring a basic understanding of classical statistical mechanics and of quantum mechanics.
Course contents summary
Introduction to Analytical Mechanics.
Statistical Mechanics of Microcanonical and Canonical Ensembles
Applications of the classical canonical ensemble.
Classical gran-canonical ensemble.
The birth of quantum mechanics (blackbody problem, photoelectrric effect, Compton effect, specific heat of solids, spectral lines, Rutherford model, Bohr model, de Broglie waves).
Wavefunction, Born interpretation,probability current, Schrodinger equation of a single particle, solution by separation of variables.
One-dimensional problems: free particle, potential well, harmonic oscillator.
Introduction to three-dimensional problems.
- Classical Mechanics in an arbitrary reference frame. Constraints, virtual displacements, generalized lagrangian coordinates. The Lagrangian of a physical systems and the Lagrange equations. Symmetries and conservation laws. Noether's theorem. Small oscillations, normal modes. The Legendre transform and the Hamiltonian. Hamilton's equations. Configuration space and phase space. Poisson brackets.
- Variational principles and Lagrange and Hamilton equations. Elements of calculus of variations. Canonical transformations. Elements of perturbation theory. Examples of relevant Lagrangians and Hamiltonians of physical systems: central forces, changed particles in an electromagnetic field. Infinite degrees of freedom: the vibrating string.
- The statistical description of a macroscopic system. Systems with many degrees of freedom and classical mechanics. Brief review of thermodynamics: extensive and intensive variables, thermodynamic potentials, Legendre transformations, response functions. Average values without dynamics: statistical ensembles and probability measures. Liouville theorem. The problems of the microscopic approach. Temporal averages and the ergodic hypothesis. Recurrence times and macroscopic variables. How and if equilibrium is reached.
- The Microcanonical Ensemble. Boltzmann entropy and its properties. Additivity. Microcanonical classical ideal gas. Gibbs paradox and correct counting. Entropy and information theory: Shannon entropy.
- The Canonical Ensemble. The partition function and the Helmotz free energy.
H. Goldstein- C. Poole - J. Safko, Meccanica Classica - Zanichelli
L.D. Laundau - E.M. Lifsits, Meccanica - Ed Riuniti
L.D. Laundau - E.M. Lifsits, Fisica Statistica, Editori Riuniti
Huang - Statistical Mechanics
Alonso-Finn - Fundamental University Physics Vol. 3 - Quantum and Statistical Physics
Eisberg - Quantum Mechanics of Atoms, Solids, Nuclei and Particles
Caldirola, Cirelli, Prosperi - Introduzione alla Fisica Teorica
Lectures and exercices
Assessment methods and criteria
Oral and written examination.