Mathematical Methods in Physics

Code:

F4025

Acronym:

F4025

Keywords
Classification	Keyword
OFICIAL	Physics

Instance: 2020/2021 - 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	11	Official Study Plan	1	-	6	42	162
			2

Teaching language

Suitable for English-speaking students

Objectives

1. To know the most usefful mathematical methods in physics, namely complex complex variable function and discrete group theory.
2. To analyze a set of problems of several areas of Physics in the perspective of the applied mathematical methods.
3. Model physics problems.

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 and intermediate 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: Complex and functional analysis in Physics
Review of the fundamental properties of complex numbers.
Complex functions of complex variable: continuity, limits, analytical functions, Cauchy-Riemann conditions, differentiation of complex functions, integral Cauchy theorem, Laurent expansion, conform maps and applications in physics (e.g. fluid dynamics, electricity and heat propagation).
Residues of complex functions: singularities, residues, analytical expansion, application to the calculation of integrals and series.
Functional spaces: fundamental properties of functional spaces, families of special functions (Legendre, Bessel, Hermite, Laguerre, etc.), generating functions, recurrence, orthogonality and normalization formulas.
Fourier transforms, Green functions and linear response theory.

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

The teaching will be the traditional one, supported in books of the specialties treated in the course.
Students will have at their disposal a set of problem sheets as a complement to the theory given.

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.