Mathematical Methods in Physics

Code:

F4025

Acronym:

F4025

Keywords
Classification	Keyword
OFICIAL	Physics

Instance: 2021/2022 - 1S

Active?	Yes
Responsible unit:	Department of Physics and Astronomy
Course/CS Responsible:	Master in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:A_ASTR	0	Plano de Estudos oficial desde_2013/14	1	-	6	42	162
			2
M:F	19	Official Study Plan	1	-	6	42	162
			2

Teaching language

Suitable for English-speaking students

Objectives

To understand the concepts and advanced mathematical techniques in the area of complex functions of complex variable, Fourier transforms, special functions and group theory.

To apply the mathematical methods in modeling and solving physical problems, for example, Electromagnetism, Optics, Fluid Mechanics, and Condensed Matter Physics.

To development skills and to get knowledge towards the development of new models and performing calculations, reading and understanding of the advanced research literature.

Learning outcomes and competences

It is intended that after the course the student can apply either theoretical or practically the varied, and often advanced, knowledge taught. It is believed that without these essential tools it is difficult to progress in more advanced and demanding subjects of Physics and Mathematics in rigor. Several books and personal notes were selected that we think are very useful.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Solid knowledge of Mathematical Analysis, intermediate Physics, Condensed Matter Physics, Quantum Mechanics and Statistical Physics, with the aim of safely moving to more advanced topics.

Program

Part 1. Group theory and applications to condensed matter physics and particle physics
Basic Mathematical Background.
Representation Theory and Basic Theorems.
Character of a Representation.
Basis Functions.
Direct productsIntroductory Application to Quantum Systems: i) Splitting of Atomic Orbitals in a Crystal Potential; ii) Application to Selection Rules and Direct Products; iii) molecular vibrations. Electronic States of Molecules and Directed Valence.Space Groups in Real Space and basic properties.  Space Groups in Reciprocal Space and Representations.Electron and Phonon Dispersion Relation.

Part 2: Advanced modeling of physical systems based on complex and functional analysis

Physics Models Based on Complex Functions of Complex Variables: Review of the fundamental properties of complex numbers, continuity, limits, analytic functions, Cauchy-Riemann conditions, differentiation of complex functions, Cauchy's integral theorem, Laurent expansion, and conformal maps. Applications in physics (e.g. fluid dynamics, electricity and heat propagation). Residues of complex functions: singularities, residues, and analytical expansion. Application to the calculus of integrals and series and applications in physics (e.g. calculus of effective sections, perturbative theory, causality, dispersion and absorption, Krammer-Kronig relations).

Physics models based on functional spaces: fundamental properties of functional spaces, families of special functions (Legendre, Bessel, Hermite, Laguerre, etc.), generating function, recurrence formulas, orthogonality and normalization. Fourier and Legendre transforms, Green functions and linear response theory. Applications in physics (e.g quantum and field theory, signal processing and dynamics of physical systems).

Mandatory literature

M. S. Dresselhaus; Group theory. ISBN: 978-3-540-32897-1
Edgar Giraldus Phillips; Functions of a complex variable with applications
George Arfken; Mathematical methods for physicists. ISBN: 0-12-059810-8

Teaching methods and learning activities

We opted for a student-centered methodology, in which the chapters of the bibliography are recommended for study are previously identified, accompanied by a questionnaire to focus on relevant concepts and theorems. In classes, the teacher will discuss the main concepts and theorems previously studied by students, illustrating with different applications. It will also resort to the discussion of problems where mathematical techniques will be applied, with special emphasis on solving physics problems. Students will be provided with a collection of exercises and problems for individual resolution, covering the topics and methods of calculation studied, having various levels of difficulty.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Frequência das aulas	49,00
Estudo autónomo	113,00
Total:	162,00

Eligibility for exams

Calculation formula of final grade

The final mark is the final exam mark.

Special assessment (TE, DA, ...)

In accordance with FCUP rules.

Classification improvement

By final exam.