43006 - Physics of Matter

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Affordable and clean energy Climate Action

Academic Year 2022/2023

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.


  • Malcolm P. Kennett, Essential Statistical Physics,Cambridge University Press
  • B.H.Bransden & C.J. Joachain, Physics of Atoms and Molecules, ISBN-13: 978-0582356924
  • Eisberg-Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, Ed. Wiley ISBN-13: 978-0471873730

Lecture notes available on Virtuale.unibo.it

Teaching methods

Frontal lectures and classroom tutorials for problem solving

Assessment methods

The final assessment is unique for the two modules and consists of two parts: 1 - written exam with 3 exercises (1 relating to the contents of the module of the first semester and 2 to the contents of the module of the second semester). The grade is 10/30 for a full year. The written test is passed only with a grade greater than or equal to 18/30 and is valid for the entire exam session in which it is passed. 2 - oral exam: the aim is to verify the student's ability to understand, elaborate and analyze the contents of the course.

Teaching tools

frontal lecture at the blackboard, slides and projector 

Office hours

See the website of Beatrice Fraboni

See the website of Cesare Franchini