## 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.
The course is also listed under the following terms Spring 2022.
• Enrolment Statistics (Spring 2025, recent)