CONDENSED MATTER PHYSICS

The course is designed to provide the students with the basic concepts to understand the physical phenomena that appear in condesed matter systems; more specifically, the course will introduce the simplified theoretical models that provide insight in the behaviour of liquids, gases and more importantly solid systems. These theoretical models provide explanations for phenomena such as current and heat conduction in metals or their magnetic properties. The course is a fundamental course for the Master Degree.

The students will learn the theoretical tools that have been developed to treat thermodynamics, and it will be shown how simplified models, such as the ideal gas model, that describes in a first approximation the behaviour of liquids and gases, can be applied to much more complex systems such as the motion of free charges in metals. After a review of thermodynamics, starting from the Drude model, which is a first classical approach to the explanation of heat and charge transport in metals, and then introducing quantum effects, the students will learn the besic theory of metals while also learning about the approach to many-body problems. Then, the simplest non-idealized quantum systems (H, He, H2 molecule) will be presented to show how quantum mechanics can give quantitatively accurate predictions and important insight. After that, the thoery of metals on a lattice will be briefly presented, and the students will learn how to treat periodic systems, their ground state and their temperature dependent behaviour.

Although not formally required, a solid understanding of the basics of statistical and quantum mechanics are very useful to be able to successfully understand the content of the lectures.

THERMODYNAMIC POTENTIALS
Equilibrium, intensive and extensive variables
Fundamental Equation of Thermodynamics
Euler and Gibbs-Duhem equation
Internal Energy
Enthalpy
Helmholtz free energy
Gibbs free energy
Grand-potential
Thermal response functions
Mechanical and Magnetic response functions
Mathematical detour: On relation on partial derivatives

INTRODUCTION TO STATISTICAL THERMODYNAMICS
Kinetic gas model
Pressure of a diluted gas
Maxwell distribution of velocities
Identical particles and symmetry of wavefunction
Example: “gas” of 2 particles and 3 states
Combinatory derivation of quantum statistics
Mathematical detour: method of Lagrange multipliers and Stirling approximation
The Method of Most Probable Distribution: Statistics of FD, BE, and MB
Particle on a Box: physical meaning of β
Chemical potential and relation to α
Thermodynamics of the MB gas

DRUDE-SOMMERFELD THEORY OF METALS
Model of Atom for a metal
The Drude model
DC Conductivity
Thermal Conductivity and Wiedeman-Franz law
Drawbacks and Sommerfeld Model
Ground state of an electron gas and Fermi Sphere
Pressure of a Fermi gas
The density of states: elementary derivation
The density of states: general derivation
Finite temperatures and specific heat
Conductivity in Sommerfeld Model and improved Wiedeman-Franz law

THEORY OF ANGULAR MOMENTUM
Angular momentum and commutation rules
The spectrum of L2 and Lz
Pauli Matrices

THE HYDROGEN ATOM
Separation of variables
The Legendre equation of the polar variable
Exact solution via the Frobenius method
Legendre polynomials
The radial Laguerre equation
Solution via the Frobenius method

THE HELIUM ATOM
Time independent perturbation theory
First order calculation of the helium ground state
Excited states: singlet and triplet states
Orthohelium and Parahelium

THE HYDROGEN MOLECULE
LCAO method
Explicit solution of the eigenvalues
Bonding and antibonding states
Physical interpretation
INTRODUCTION TO CRYSTAL STRUCTURES
Shortcomings of free electron theory
Bravais lattice
Reciprocal lattice
Examples

BAND THEORY
Motivations
Bloch theorem
Born-von Karman periodic boundary conditions
Mathematical detour: periodic functions
Central equation for electron in a periodic potential
Nearly free electrons model
Band gaps and Brillouin zones
Bragg scattering and Bragg planes

CLASSICAL THEORY OF HARMONIC CRYSTALS
Shortcomings of fixed ions theory
Classical equipartition function and Doulong-Petit theory
One-dimensional crystal and harmonic approximation
Normal modes of a biatomic molecule
Normal modes of a harmonic crystal
Normal modes of a harmonic crystal with basis

QUANTUM THEORY OF HARMONIC CRYSTALS
Quantum theory of harmonic oscillator
Einstein theory of specific heat in solids
Debye theory of specific heat in solids
Estimates of Debye temperature and wavevector and physical interpretation

Solid State Physics
N. W. Ashcroft e N. D. Mermin: Solid State Physics
M. L. Cohen, and S.G. Louie: Fundamentals of Condensed Matter Physics
C. Kittel: Introduction to Solid State Physics

Atomic and Molecular Physics
C. Cohen, B. Diu, F. Laloe: Quantum Mechanics Vol 1 e 2
R. Feynman, R. Leighton e M. Sands: La Fisica di Feynman Vol III
M. Blinder: Introduction to Quantum Mechanics
F. Reif: Fundamental of Statistical and Thermal Physics

Marks will be given with an oral examination. The questions will be designed to test the level of understanding of the physical concepts presented during the lectures.

Frontal lectures with digital aids (touch tablet and screen projector)

English

oral