# PřF:FA040 Adv. math. meth. in th. phys. - Course Information

## FA040 Advanced mathematical methods in theoretical physics

**Faculty of Science**

Spring 2025

**Extent and Intensity**- 1/1/0. 3 credit(s). Type of Completion: k (colloquium).
**Teacher(s)**- prof. Mgr. Tomáš Tyc, Ph.D. (lecturer)

Mgr. Darek Cidlinský (seminar tutor)

prof. Mgr. Tomáš Tyc, Ph.D. (seminar tutor) **Guaranteed by**- prof. Mgr. Tomáš Tyc, Ph.D.

Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science

Contact Person: prof. Mgr. Tomáš Tyc, Ph.D.

Supplier department: Department of Theoretical Physics and Astrophysics – Physics Section – Faculty of Science **Prerequisites**- Knowledge of mathematical methods at the level of Mgr. study of Physics.
**Course Enrolment Limitations**- The course is offered to students of any study field.
**Course objectives**- The main goal of the course is to provide students with tools useful in many areas of theoretical and practical physics, with the emphasis on the methods of group theory.
**Learning outcomes**- After finishing the course, the students will be able to:

- find generators and classify representations of the most common groups used in physics

- derive equations for various special functions and orthogonal polynomials

- be able to apply algebraic approach to solving various problems in quantum mechanics

- perform calculations with Dirac delta functions and Fourier transformations

- use conformal mapping and its properties for solving a variety of problems in physics

- work with spherical geometry, stereographic projection, and geometry of the hyperbolic plane and the pseudosphere **Syllabus****Introduction to group theory:**- Groups and their properties; discrete groups, the symmetric group; Lie groups, their examples; Group generators, Lie algebras; Group representations; Reducible and irreducible representations, examples; Applications in physics, angular momentum
**Special functions and orthogonal polynomials:**- Legendre polynomials and their relation to representations of the group SU(2), spherical harmonics; Laplace operator in polar coordinates, Bessel and Hankel functions; Harmonic oscillator and Hermite polynomials; 2D isotropic harmonic oscillator and Laguerre polynomials
**Integral transformations:**- Fourier transformation and its applications; Position and momentum representations; Properties of the Dirac delta function and using it in calculations
**Selected chapters from complex analysis:**- Möbius transforms; Conformal mappings and their applications in physics; Non-Euclidean geometry, stereographic projection, geometry of the sphere; Hyperbolic plane and pseudosphere

**Literature**- I.M. Gelfand, R.A. Minlos, Z. Ya Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications
- N. J. Vilenkin, Special functions and the theory of group representations
- NEEDHAM, Tristan.
*Visual complex analysis*. 1st pub. Oxford: Clarendon Press, 1997, xxiii, 592. ISBN 0198534469. info

**Teaching methods**- The theoretical part will have the form of a lecture. The tutorials will not be given separately but rather will interlace with the lecture: for example, after a certain topic is explained, the students will be asked to make some related calculation on the blackboard. Group discussions will also be encouraged.
**Assessment methods**- The course will be concluded by a colloquium, i.e., a discussion with the student that will assess whether he/she has understood the course and has a good overview over the topics. The presence at the lectures jonied with tutorials will also be required, with the maximum of three absences.
**Language of instruction**- English
**Further comments (probably available only in Czech)**- The course is taught once in two years.

The course is taught: every week.

General note: L.

- Enrolment Statistics (Spring 2025, recent)
- Permalink: https://is.muni.cz/course/sci/spring2025/FA040