## Admission Requirements

Quantum Theory a

Bachelor of Physics with an introduction to solid state physics and (preferably) some knowledge on semiconductors and electron bands

## Description

The course gives an introduction into the theory of quantum phenomena in condensed matter systems.

With the help of the second quantization approach, perturbation theory and the mean-field theory the course introduces a number of fundamental concepts such as long-range order, spontaneous symmetry breaking, elementary, collective and topological excitations. These general concepts are illustrated on a range of archetypal examples such as crystalline dielectric solid, superfluid, normal metal and superconductor.

The detailed list of topics includes

Second quantization of bosonic/fermionic fields

Elementary excitations in harmonic crystals

Thermodynamics of harmonic crystals

Effects of anharmonicity in real crystals

Probing elementary excitations with neutron scattering and other techniques

Elements of kinetic theory

Linear response theory and the Kubo formula

Superfluidity, the two-fluid model, elementary excitations and the Landau criterion.

Bose-Einstein condensation

Bogoliubov's theory of a superfluid

Condensate depletion

The Gross–Pitaevskii equation

The linear sigma-model

Topological excitations in a superfluid

Thermodynamic properties of a Fermi gas

Magnetic properties of a Fermi gas

The Hartree-Fock approximation

Renormalization of the parameters of a Fermi liquid in the Hartree-Fock approximation

The Landau Fermi-Liquid theory

The RPA approximation

Collective excitations in a Fermi liquid

Superfluidity

The Landau-Ginzburg theory of a superfluid

The electron-phonon interaction

The Cooper instability

The BCS theory

## Course objectives

The course will provide students with a working knowledge of the mathematical

framework of quantum many-body theory, including the second quantization formalism, quantum statistical mechanics, linear response theory, and the mean-field theory.

The course will also familiarize the students with the key ideas of quantum liquid phenomenology including spontaneous symmetry breaking, long-range order, elementary excitations, hydrodynamics, and the effective low-energy Hamiltonians.

At the end of the course you will be able to

Construct second-quantized models of quantum many-body systems

Calculate thermodynamic properties of model systems

Calculate linear response functions (e.g. magnetic susceptibility) of model systems

Describe elementary excitations of a model system

Use perturbation theory in a many-body system

Calculate scattering cross sections of elementary excitations and relaxation rates

Calculate form factors for scattering experiments (e.g. neutron scattering)

Apply mean-field theory to interacting systems of bosons and fermions

Use the semiclassical theory for the long-range dynamics of a quantum fluid

Construct topological excitations of a quantum fluid

Use the random phase approximation

Calculate the properties of a superconductor within the Landau-Ginzburg theory

Derive and solve the BSC equation for the superconducting gap

## Generic skills (soft skills)

## Timetable

## Mode of instruction

Lectures and tutorials

## Course load

## Assessment method

written examination with short questions

## Blackboard

Blackboard will be used for the provision of lecture notes, distribution of home assignment worksheets, and announcements.

To have access to Blackboard you need a ULCN-account.Blackboard UL

## Reading list A set of lecture notes prepared by the lecturer and

R. Feynman, Statistical Mechanics: A Set of Lectures

C. Kittel, Quantum Theory of Solids

D. Pines, Elementary Excitations in Solids

D.R. Tilley and J Tilley, Superuidity and Superconductivity

P. Nozieres and D. Pines, Theory of Quantum Liquids

M. Tinkham “Introduction to superconductivity”

## Contact

Contactdetails Teacher(s):Dr. Vadim Cheianov