# QUANTUM THEORY OF SOLIDS

## Learning outcomes of the course unit

The goal of this course is to provide the students with both a conceptual and quantitative understanding of solids.

The course is well suited for students who want to achieve a fundamental understanding of the connection between different systems in condensed matter physics, such as metals, semiconductors, superconductors, and strongly correlated systems.

They should obtain a good insight into modern quantum theory of solids, and further, through chosen examples, obtain a deeper understanding of key concepts within condensed matter physics, such as phase transitions and correlations.

After completed course, the students should be able to: formulate the many particle problem in second quantized version, use theoretical methods for the many body problem to solve problems covered in the course, give an account of the problems in the area that are treated in the course.

## Prerequisites

The course consolidates many courses from your degree, as it touches on almost all of the physics courses that you have taken in undergraduate program including electricity and magnetism, thermodynamics, quantum mechanics, classical mechanics, as well as solid state physics at introductory level.

In addition, it requires a good working knowledge of Quantum Mechanics and Statistical Physics at a graduate level.

## Course contents summary

The course gives an introduction into the theory of collective quantum phenomena in condensed matter systems. Target audience is made up from students in physics or material science who are interested in understanding the theoretical approach to Solid State Physics, while maintaining contact to the experimental facts.

Topics include: theory of linear response, second quantization, the electron gas, one electron theory, lattice dynamics, scattering of particles by crystals, electron-phonon interaction, superconductivity, light-matter interactions.

## Course contents

1. The Solid as a Many-Particle Problem.

Spontaneously broken symmetry; The Hamiltonian of a solid; Born-Oppenheimer approximation; second quantization for bosons and fermions (just a quick summary since this topic has already been developed in the 1st semester in the “Statisical Physics” course); theory of linear response, Kubo’s formulas and response functions.

2. Free electrons without interaction; the electron gas in magnetic fields; occupation number representation; Hartree-Fock approximation; perturbation theory; the dielectric function.

3. One Electron Theory.

Density functional theory; the Hohenberg-Kohn theorem and the Kohn-Sham formulation; Bloch electrons and band structure; almost free electrons, the core states and pseudo-potentials; LCAO and tight binding approximation; effective mass approximation and the dynamics of Bloch electrons; subbands in semiconductor quantum structures.

4. Lattice Dynamics .

Harmonic approximation and normal coordinates; phonons and occupation number representation; acoustic and optical modes; density of states and the Debye approximation; beyond the harmonic approximation and phonon interactions; the crystal lattice at finite temperatures.

5. Scattering of Particles by Crystals.

Elastic scattering of X-rays and the Thompson approximation; inelastic scattering of particles and phonon spectra; quantum theory of elastic and inelastic neutron scattering and the dynamic structure factor.

6. Electron-Phonon Interactions.

Coupling mechanisms and scattering processes (lifetime, relaxations); the Frölich Hamiltonian; phonon frequencies and the Kohn anomaly; the Peierls transition; polarons.

7. Superconductivity.

The superconducting state; the BCS Hamiltonian; the ground state function and the energy gap; the transition temperature; the Meissner effect; tunnelling experiments; flux quantization and the Josephson effect; the Ginzburg-Landau equations; high temperature superconductivity.

8. Light-Matter Interactions.

Single particle approximation; excitons; polaritons; light scattering; interband dynamics.

## Recommended readings

Recommended reference textbooks:

U. Rössler - Solid State Theory: An Introduction, Springer 2009

This is a Solid-State textbook for graduate students of physics and material sciences. Whilst covering the traditional topics of older textbooks, it also takes up new developments in theoretical concepts and materials that are connected with such breakthroughs as the quantum-Hall effects, the high-Tc superconductors, and the low-dimensional systems realized in solids. Thus besides providing the fundamental concepts to describe the physics of the electrons and ions comprising the solid, including their interactions, the book casts a bridge to the experimental facts and gives the reader an insight into current research fields. A compilation of problems makes the book especially valuable to students.

P.L. Taylor & O. Heinonen - A Quantum Approach to Condensed Matter Physics, Cambridge University Press 2002

This is a reader-friendly introduction to Solid State Theory: Taylor and Heinonen describe the methods for performing calculations and making predictions of some of the many complex phenomena that occur in solids and quantum liquids. Their book aimed at advanced undergraduates and beginning graduate students, leads the reader from the fundamental behaviour of electrons and atoms in solids to the most recently explored manifestations of the quantum nature of condensed matter.

G. Grosso & G.Pastori Parravicini - Solid State Physics 2nd Edition, Academic Press 2014

This book fills the gap between the active field of research and the concepts traditionally taught in solid-state courses. The style is tutorial, simple, and completely self-contained. Examples are an integral part of the text, and they are carefully designed to apply the fundamental principles illustrated in the text to currently active topics of research.

## Teaching methods

Lectures, colloquia, or guided self-study, depending on the number of students.

There will be homework assignments about every four weeks.

## Assessment methods and criteria

There will be a final written exam (no mid-term exams).

Students should prepare a short oral presentation on a topic of their choice relating to the course

The grading will be based on:

• homework assignments 40%

• oral presentation 20%

• final written exam 40%