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Course, academic year 2023/2024
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Asymptotic Methods in Physics - NBCM184
Title: Asymptotické metody ve fyzice
Guaranteed by: Department of Chemical Physics and Optics (32-KCHFO)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Ing. Lucie Augustovičová, Ph.D.
Annotation -
Convergent versus asymptotic expansions. Asymptotic relations and expansions - properties, algebraic and analytical operations with them. Various methods for asymptotic evaluation of parametric integrals. Applications in problems of mathematical physics.
Last update: Kapsa Vojtěch, RNDr., CSc. (14.01.2020)
Aim of the course -

The aim of this course is to deepen and broaden knowledge of approximate methods with applications in physics.

Last update: Kapsa Vojtěch, RNDr., CSc. (17.01.2020)
Course completion requirements -

The course credit is awarded for participation and activity in exercises. Insufficient participation cannot be compensated in any other way.

Course credit is a condition of admission to the exam.

The exam is oral and the requirements correspond to the syllabus of the subject in the range that was presented at the lecture.

Last update: Kapsa Vojtěch, RNDr., CSc. (17.01.2020)
Literature -
  • Bender, A. S. Orszag, Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, Springer-Verlag New York, 1999.
  • M. V. Karasev, (Mikhail Vladimirovich), Asymptotic Methods for Wave and Quantum Problems, Providence, Rhode Island: American Mathematical Society, 2003.
  • E. T. Copson, Asymptotic Expansions, Cambridge University Press, 1965.
  • Ivan Avramidi, Lecture Notes on Asymptotic Expansion, New Mexico Institute of Mining and Technology, 2000

Last update: Kapsa Vojtěch, RNDr., CSc. (14.01.2020)
Teaching methods -

Lectures and exercises.

Exercises contain examples of asymptotic expansions and their properties, basic properties of asymptotic expansions and algebraic operations with them, asymptotics of Laplace-type integrals, applications of Watson's lemma, application of Laplace's method in symbolic programming etc.

Last update: Kapsa Vojtěch, RNDr., CSc. (20.01.2020)
Syllabus -

1. Convergent versus asymptotic expansions, Landau notation

2. Asymptotic expansion of functions, Stirling expansion of the Gamma function

3. Asymptotic series, Padé approximants, continued fractions

4. Asymptotic expansion of Laplace integrals, Watson's lemma

5. Laplace's method

6. Method of steepest descents for one-dimensional integrals: Example of asymptotic expansion of Bessel functions

7. Stationary phase method (in optics), eikonal, diffraction

8. Method of steepest descents for multidimensional integrals, Feynman diagrams, demonstration of one and two-loops calculations

9. WKB method in quantum mechanics, the anharmonic oscillator.

10. Solution of nonlinear differential equations by WKB method; application to fluid mechanics

11. Theory of a scalar quantum field, one and two loop calculation of effective potential

12. Introduction to renormalization theory

Last update: Kapsa Vojtěch, RNDr., CSc. (17.01.2020)
 
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