## 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