34270 - Elements of Structure of the Matter

Learning outcomes

At the end of the course, the student has got basic knowledge of mechanics and Hamiltonian phase space, ergodic hypothesis, the time averages and averages in phase space. In particular, the student has detailed knowledge about the method of Boltzmann entropy maximization, systems microcanonici and canonical partition functions for discrete spectra and continuum limit, the canonical distribution. The student will acquire the basic knowledge of: indistinguishability,  not degenerate gas, equation of state and thermodynamic functions for ideal gas, Gibbs paradox, an ideal gas in an external field and the barometric formula, weakly ionized atomic gas and the Saha formula, gas molecular and thermal equilibrium of chemical reactions. Moreover, at the end of the course, the student possesses the basic knowledge about the solutions of the Schrödinger equation in a time-independent one-dimensional systems, eigenvalues ​​and eigenfunctions for the problem of Schrödinger atom idrogenoide, expectation values ​​and symmetry properties, rules selection for transitions between states in the approximation of the electric dipole. In particular, the student is able to calculate the magnetic dipole moments, spin and magnetic moment of spin, total angular momentum and total magnetic moment. The student will acquire  the fundamental knowledge of Stern-Gerlach experiment, the fine structure of hydrogenic states, the normal and anomalous Zeeman effect, the selection rules for radiative transitions, the states of multi-electronic atoms, the Pauli principle, the algorithm of the Hartree, exchange forces, the excitation X-rays, the specific heat of a solid, the laser, the LCAO method, the molecular orbitals, the band theory, crystalline structures, molecules and molecular spectra.

Course contents

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.

Atomic Models

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

One-electron atoms

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 atoms

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 cental 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 spectra.

Crystalline solids

Introduction to the band theory in solids; Bloch theorem; Insulating, semiconducting and conducting materials.

Readings/Bibliography

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, Ed. Wiley ISBN-13: 978-0471873730

Teaching methods

The topics will be presented in such a way to stimulate the ability to identify similarities/differences among the various models, the experimental results and theories presented.

Assessment methods

Assessment is through a written plus an oral test. The written test (1:30h) consists in 2 exercises and one question (maximum length of the answer is half a page) to assess the student's knowledge of the topics discussed during the lectures. The written test is passed only with a grade equal or above 18/30 and the oral test has to be  will present a difficulty level similar to that of  the exercises discussed during the lectures.

Teaching tools

Blackboard, overhead projection

Office hours

See the website of Beatrice Fraboni