43006 - Physics of Matter

Learning outcomes

At the end of the class the student will have acquired: a) fundamental principles of statistical physics, 

Course contents

Module I semester:

Probability Theory:Real random variables. Single-variable probability function. Probability distributions. Dirac’s Delta function. Changes of variables. The Gaussian. Characteristic functions. Many-variables distributions. Correlations. Sum of independent variables and Central Limit Theorem.

Statistical Thermodynamics: from Dynamics to Thermodynamics:Empirical thermodynamics: the 3 Principles. Heat and Temperature. From Dynamics to Thermodynamics: heat exchanges as generalized scattering events. Thermodynamic functions as time averages. Liouville theorems for Hamiltonian classical systems. Micro-canonic, Canonic. Grand-canonic systems. Ergodic systems. Partition of a micro-canonic system into canonic sub-systems. Thermodynamic limit. (In)Distinguishability. Boltzmann’s method. Derivation of Temperature and Entropy. Boltzmann Principle as a theorem.

Non degenerate systems:Distinguishable harmonic oscillators. Non-degeracy limit. Continuum limit. The Perfect Gas. Equipartition Theorem. Maxwell-Boltzmann Distribution. Perfect Gases in the gravitational field: Barometric Formula. Deriving Archimede’s Principle. Atomic and molecular gases. Thermal equilibrium of chemical reactions: the Mass Action Law and Saha Formula.

Degenerate Gases:Bosons and Fermions. Chemical potential. Continuum limit for Bosons: Bose-Einstein Condensation. Condensation temperature. Massless bosons and gases of quantum oscillators. Black Body and Planck Formula. Degenerate Fermions and Fermi level. Insulators and conductors from a band spectrum picture. Sommerfeld expansions for conducters. Effective Fermions.

Module II semester:

Atomic Models: Atomic spectroscopy, Thomson’s model, Rutherford’s model, Bohr’s model, Franck-Hertz experiment, Sommerfeld model

One-electron atom (H): The Schroedinger equation and its solution for the Hydrogen atom: energy levels and eigenfunctions of the bound states; radial distribution density. Orbital angular momentum and magnetic dipole moment; Stern-Gerlach experiment; Spin, Spin-orbit interaction. Dirac equation, perturbative solutions; Fine structure; Lamb shift and hyperfine structure. Selection rules and transition rates; Spectral line width and shapes.

Two-electron atom (He): The Schroedinger equation for two-electron atoms: ortho and para states. Spin wave functions and the Pauli exclusion principle. Energy level scheme for two-electron atoms. Ground state and excited states; Coulomb integral and exchange integral.

Many-electron atoms: The central field approximation; Hartree-Fock model and Slater determinants. The periodic table of the elements. X-ray spectra, Moseley’s law. Corrections to the central field approximation: L-S coupling and j-j coupling. Zeeman effect.

Molecules: Molecular structures. Ionic and covalent bond. The H2+ ion; Bonding and antibonding orbitals; Born-Oppenheimer approximation, LCAO method. Molecular roto-vibrational spectra (harmonic and anharmonic approximation)

Crystalline solids: Introduction to the band theory in solids; Crystalline and periodic structures; Bloch theorem, electrons in a solid; electron wave function in a lattice; Insulating, semiconducting and conducting materials.

Office hours

See the website of Beatrice Fraboni

See the website of Cesare Franchini