Mathematical Methods in Physics Field of study: Physics
Programme code: W4-S2FZA22.2.2021

Module name: Mathematical Methods in Physics
Module code: W4-2F-17-15
Programme code: W4-S2FZA22.2.2021
Semester: winter semester 2021/2022
Language of instruction: English
Form of verification: exam
ECTS credits: 5
Description:
The lecture includes a coherent and uniform presentation of elements of the theory with justifications and many examples derived from physics and engineering within the following topics: 1. Elements of distribution theory: basic concepts, differentiation of distribution, the Dirac delta, and related distributions, the principal value of the integral; operations on distributions; Sochocki formulas, the convolution of distributions and their Fourier transform. 2. Green's functions of differential operators: boundary issues, related to eigenvalue problem; examples coming from physics and engineering (e.g. Sturm Liouville systems). 3. Elements of Hilbert space theory: basic concepts and examples; orthonormal and Schauder bases; unitary and self-adjoint operators; spectra and eigenvalues; subtleties of the formalism of quantum theory. 4. Fourier series and their properties. 5. Integral transforms; Fourier and Laplace transform and their properties. 6. Elements of signal analysis. The classes and seminars are devoted to solving selected examples and explaining theories in specific physical situations. Students participate in deriving and discussing some formulas and examples from lectures, as well as the discussions of the significance of the discussed formalisms in various physical problems. As part of the student's work the student: 1. strives to consolidate acquired knowledge based on lecture notes and supplementary literature; 2. improves the mathematical skills necessary to solve physical problems; 3. tries to accomplish the tasks proposed by the lecturer. The exam is compulsory.
Prerequisites:
Knowledge of basic problems of mathematical analysis and algebra (mathematics courses at first-cycle studies).
Key reading:
P. Blanchard, E. Brünning, Mathematical Methods in Physics, Birkhäuser (Springer) 2015 G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, Academic Press 2001 (or later editions) E. N. Economou, Green’s Functions in Quantum Physics, Springer 2006 H-W. Steeb, Hilbert Spaces, Wavelets, Generalized Function and Modern Quantum Mechanics, Springer 1998 P. Sołtan, A Primer on Hilbert Space Operators (this book is also available in Polish) W. Mlak, Hilbert Spaces and Operator Theory, PWN 1991 (this book is also available in Polish) Ø. Ryan, Linear Algebra, Signal Processing and Wavelets – A Unified Approach, Springer 2019. T. Olson, Applied Fourier Analysis, Springer 2017 A. Boggess, F. J . Narcowich, A first Course in Wavelets and Fourier Analysis, Wiley & Sons 2009A. Boggess, F. J . Narcowich, A first Course in Wavelets and Fourier Analysis, Wiley & Sons 2009
Learning outcome of the module Codes of the learning outcomes of the programme to which the learning outcome of the module is related [level of competence: scale 1-5]
understanding the civilization meaning of differential and integral calculus and its role in physics; [2F_15_1]
 KF_W01 [4/5] KF_U01 [4/5]
the student has a good theoretical and practical intuition related to mathematical analysis; is able to perform basic calculations; [2F_15_2]
 KF_W02 [4/5] KF_U02 [4/5]
understands the meaning and can give examples of the physical application of differential equations in physics and technology; [2F_15_3]
 KF_U01 [3/5] KF_U02 [3/5]
understands and is able to perform simple calculations on Hilbert spaces; [2F_15_4]
 KF_W05 [3/5] KF_U03 [3/5]
understands the need to use the distribution theory tools in various branches of physics - can calculate the Fourier transform, convolution, derivatives, e.g. for the Dirac delta. [2F_15_5]
 KF_W05 [3/5] KF_U03 [3/5]
knows the concept of Fourier analysis and its applications in various fields of physics. [2F_15_6]
 KF_W05 [3/5] KF_U03 [3/5]
The student understands (through examples) the need to develop mathematical formalism in order to better describe and understand the physical world. [2F_15_7]
 KF_W01 [4/5]
Type Description Codes of the learning outcomes of the module to which assessment is related
colloquium [2F_15_w_1]
Optional verification method; the date of the colloquium or written test announced to students two weeks earlier; tasks of a similar type to the tasks solved during the seminar
 2F_15_2 2F_15_3 2F_15_4 2F_15_5
activity in class [2F_15_w_2]
problem solving and discussion of the discussed problems (basic method)
 2F_15_1 2F_15_6 2F_15_7
written exam (or oral exam) [2F_15_w_3]
the condition for taking the exam is passing the conservatory; scope of the material - all issues discussed during the lectures
 2F_15_1 2F_15_4 2F_15_5 2F_15_6
Form of teaching Student's own work Assessment of the learning outcomes
Type Description (including teaching methods) Number of hours Description Number of hours
lecture [2F_15_fs_1]
lecture of selected basic issues with the use of audiovisual aids
 30
supplementary reading, work with the textbook
 40 written exam (or oral exam) [2F_15_w_3]
discussion classes [2F_15_fs_2]
solving tasks at the blackboard
 30
supplementary reading
 40 colloquium [2F_15_w_1] activity in class [2F_15_w_2]
Attachments
Module description (PDF)
Information concerning module syllabuses might be changed during studies.
Syllabuses (USOSweb)
Semester Module Language of instruction
(no information given)