Course detail

# Quantum and Statistical Physics

FSI-TQSAcad. year: 2019/2020

The course represents the third and final part of the basic course of theoretical physics.

It is concerned with:

A)QUANTUM MECHANICS. Following a discussion on the motivation for a quantum theory, the basic ingredients of the quantum-mechanical description of the microparticle are introduced: the probability amplitude and wavefunction, mathematical tools, postulates. Then the following issues are covered: one dimensional problems (potential barriers and wells – tunnelling and energy quantisation), harmonic oscillator (creation and annihilation operators), three dimensional problems (angular momentum, hydrogen atom), spin, systems of identical particles (fermions and bosons), approximate methods (time-independent and time-dependent perturbation theories, variational method). Finally advanced topics, such as density matrix, entangled states, Bell inequalities, quantum cryptography, teleportation, cloning, quantum computers, are mentioned.

B)STATISTICAL PHYSICS. Using the notion of accessible quantum states, the entropy, temperature and chemical potential are introduced and Gibbs/Boltzmann factors are deduced. Next, it is shown how to extract thermodynamic information from the partition function. Further, the Fermi-Dirac and Bose-Einstein distributions are derived. The approach unites statistical mechanics and thermodynamics. The final part aimed at applications covers the following topics: the mono- and poly-atomic ideal gas, Fermi gas in metals, black-body radiation, Bose-Einstein condensation in Bose gas.

Supervisor

Department

Learning outcomes of the course unit

Students are acquainted with formal structure of quantum mechanics and basics principles of statistical physics, and also trained to apply the knowledge gained to solve simple problems, in particular related to atomic and solid state physics.

Prerequisites

Basics of theoretical mechanics, in particular Hamiltonian dynamics. Basics of atomic physics. Particle character of radiation and wave nature of particles. The kinetic theory of gases and basics of thermodynamics (heat and the first law of thermodynamics, entropy and the second law of thermodynamics, Carnot cycle).
MATHEMATICS: Basic knowledge of functional analysis (Hilbert space, orthogonal systems of functions).

**compulsory prerequisite**

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Griffiths, David Jeffrey. Introduction to quantum mechanics. Englewood Cliffs : Prentice Hall, 1995. (EN)

Griffiths, David Jeffrey. Introduction to quantum mechanics. Englewood Cliffs : Prentice Hall, 1995. (EN)

Landau, Lev Davidovič - Lifšic, Jevgenij Michajlovič. Úvod do teoretickej fyziky. 2, Kvantová mechanika. 1. vyd. Bratislava : Alfa, 1982. (SK)

Kittel, Ch., Kroemer H. : Thermal Physics. W. H. Freeman, New York 2000 (EN)

FEYNMAN, R.P.-LEIGHTON, R.B.-SANDS, M.: Feynmanovy přednášky z fyziky, Fragment, 2001 (CS)

Kittel, Ch., Kroemer H. : Thermal Physics. W. H. Freeman, New York 2000 (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

The exam is combined (written and oral). Students have to demonstrate their knowledge of basic principles of quantum mechanics and statistical physics, and their ability to apply the knowledge gained to solve simple problems.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The objective of the course is to provide students with basics ideas and methods of quantum and statistical physics in order to be capable of understanding microscopical nature of matter and principles, which the advanced materials technologies are based on.

Specification of controlled education, way of implementation and compensation for absences

Attendance at seminars is required and recorded by the tutor. Missed seminars have to be compensated.

#### Type of course unit

Lecture

52 hours, optionally

Teacher / Lecturer

Syllabus

QUANTUM MECHANICS

1. Motivation for quantum mechanics

2. Analogy geometrical vs. wave optics ; classical vs. quantum mechanics

3. The probability amplitude and wavefuction

4. Quantum states and operators: Hilbert space, observables and Hermitian operators

5. Position representation: the position operators, Dirac delta function, the momentum operator, commutation relation for position and momentum, connection between the position and momentum representations

6. Generalized uncertainty principle

7. Schrödingers equation: Hamiltonian, stationary states, evolution of non-stationary states

8. One-dimensional problems: potential barriers and wells - tunneling and quantisation

9. Harmonic oscillator: creation and annihilation operators, application: photons, phonons

10. Angular momentum in quantum mechanics: integral and half-integral angular momentum, spin 1/2

11. Hydrogen atom

12. Approximate methods: time-independent and time-dependent perturbation theory, Fermi golden rule, variational principle

13. Identical particles: bosons and fermions

14. Advances topics: density matrix, entangled states, Bell inequalities, Greenberger-Horne-Zeilinger states. Note on quantum cryptography, teleportation, cloning, and quantum computers.

STATISTICAL PHYSICS

1. Statistical thermodynamics

2. Two systems in thermal contact: entropy and temperature, two systems in diffusive contact: the chemical potential

3. Gibbs and Boltzmann factors: partition functions, average values of physical quantities (energy, number of particles)

4. Bridge between statistical physics and thermodynamics, thermodynamic identity

5. Quantum and classical ideal gas: indistinguishable principle, fermions: Fermi-Dirac distribution, bosons: Bose-Einstein disrtribution, classical limit: Boltzmann distribution

6. Applications

6.1. Classical ideal gas (the monoatomic and polyatomic gas, heat capacity, equation of state, kinetic theory, Maxwell distribution of velocities, electric/magnetic dipoles in electric/magnetic field

6.2. Fermi gas in metals (density of states, Fermi energy, heat capacity, equation of state)

6.3. Boson physics (black-body radiation, Planck radiation law, the Stefan-Boltzmann law, the equation of state of photon gas, lattice vibrations: phonons in solids, Bose-Einstein condensation – superfluidity)

7. Thermodynamic potentials (energy, Helmholtz free energy, Gibbs free energy, heat function, grand potential)

8. Boltzmann transport equation: the electrical conductivity of metals

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Schedule of tutorials: http://physics.fme.vutbr.cz/ufi.php?Action=&Id=975

eLearning

**eLearning:** currently opened course