TFNSP0003 Mathematical Methods in Physics

Institute of physics in Opava
winter 2021
Extent and Intensity
3/2/0. 7 credit(s). Type of Completion: zk (examination).
Teacher(s)
RNDr. Filip Blaschke, Ph.D. (lecturer)
RNDr. Martin Blaschke, Ph.D. (seminar tutor)
Guaranteed by
RNDr. Filip Blaschke, Ph.D.
Institute of physics in Opava
Timetable
Mon 8:55–11:20 309
  • Timetable of Seminar Groups:
TFNSP0003/01: Mon 13:55–15:30 309, M. Blaschke
Prerequisites (in Czech)
(FAKULTA(FU) && TYP_STUDIA(N))
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The goal of this course is to introduce students to the most important mathematical tools used in physics. These fall into three broad categories: i) Complex analysis, ii) discrete calculus and iii) Ordinary differential equations. The emphasis is placed on practical calculations while rigorous presentation ala definition-theorem-proof is deemphasized.
Learning outcomes
By completing the course the student should be capable of:
- basic orientation in complex analysis;
- solve proper real integrals as contour integrals in the complex plane;
- find exact solutions to discrete and recurrent equations;
- find exact solutions to differential equations;
- find approximations to solutions of differential equations via perturbation and asymptotic methods;
- find approximations to energy levels and asymptotic behavior of the wave functions for a general one-dimensional Schroedinger equation.
Syllabus
  • The main topics are:
    • Introduction to discrete calculus. The primitive function theorem, inverse operator to indefinite summation. Discrete product and its inverse operator. Finding solutions to simple recurrence equations.
    • Binomial numbers and related identities. Integral representation. Stirling numbers and other special sequences.
    • Introduction to complex analysis. Analytic function, Riemann-Cauchy conditions. Cauchy's theorem.
    • Classification of singularities in the complex plane. Laurent series and Residue theorem.
    • Computation of proper integrals via techniques of complex analysis.
    • Solving general linear differential equations. Classification of singular points. Taylor series. Airy equation.
    • Frobenius series as a solution to the differential equation in its regular-singular point. Modified Bessel equation.
    • Introduction to asymptotic methods. Definition of the asymptotic relation. Method of dominant balance.
    • Perturbation series and its convergence. Divergent series. Pade series.
    • WKB method. An approximate solution to inhomogeneous differential equations.
    • Sturm-Liouville problem. Asymptotic solution of the Schroedinger equation with a single turning point. Global approximation.
    • WKB approximation of the Schroedinger equation with two turning points and semiclassical quantization condition.
    • Exact solution to selected Schroedinger equations.
Literature
    recommended literature
  • Bender, I. C. M., Orszag, S. A. Advanced Mathematical Methods for Scientists and Engineers, Springer, 1999
  • GRAHAM, R., KNUTH, D., PATASHNIK, O. Concrete Mathematics. Addison-Wesley, New York, 1992. info
  • REKTORYS, K. a spol. Přehled užité matematiky 2. Praha : Prometheus, 2000. ISBN 80-7196-181-7. info
Teaching methods
Live lectures in front of the blackboard or presentation in front of a projector. Showcasing of approximative solutions in Mathematica software.
Assessment methods
The written examination followed by a discussion with the teacher.
Language of instruction
Czech
Further Comments
Study Materials
The course can also be completed outside the examination period.
The course is taught annually.
The course is also listed under the following terms winter 2020, winter 2022, winter 2023, winter 2024.
  • Enrolment Statistics (winter 2021, recent)
  • Permalink: https://is.slu.cz/course/fu/winter2021/TFNSP0003