Guide in PDF format

Advanced Mathematical Methods

Code: 107624
Credits: 6

2026/2027

Degree programme	Type	Course
Physics	OB	2

Contact lecturer

Name :: Santiago Peris Rodriguez
Email :: santiago.peris@uab.cat

Group languages

You can consult this information at the end of the document.

Prerequisites

It is advisable to have studied the following subjects:

Calculus in one variable

Vector Calculus

Differential equations

Objectives

This subject introduces some basic mathematical concepts

needed in physics in general, and in physics / Quantum mechanics

and field theories, in particular. It is intended that the student

achieve the understanding of the concepts of Hilbert space, operators, distributions

and, especially, groups. It wants to give an integrative vision

of concepts that appear in different fields in physics. At the same time,

the student will have to acquire the capacity to apply them with agility

for different types of problems.

Learning outcomes

CM23 (Adapt the advanced mathematical strategy when addressing a complex problem determined from an analytical point of view.) Adapt the advanced mathematical strategy when addressing a complex problem determined from an analytical point of view.
KM22 (Identify the advanced concepts of vector and subspace space, bilinear and scalar and tensor product, and the methodology of endomorphism diagonalisation.) Identify the advanced concepts of vector and subspace space, bilinear and scalar and tensor product, and the methodology of endomorphism diagonalisation.
KM23 (Describe the basic concepts of calculus and analysis and the different methods of solving differential equations in their different typologies.) Describe the basic concepts of calculus and analysis and the different methods of solving differential equations in their different typologies.
KM24 (Identify the different types of integral transformations, the probability spaces of events, the foundations of probability theory and the basic concepts of statistical data analysis.) Identify the different types of integral transformations, the probability spaces of events, the foundations of probability theory and the basic concepts of statistical data analysis.
KM25 (Describe a Hilbert space, a linear operator and the theorems of eigenvalues and eigenvectors, the group theory SO (3), SU (2) and Lie algebra.) Describe a Hilbert space, a linear operator and the theorems of eigenvalues and eigenvectors, the group theory SO (3), SU (2) and Lie algebra.
SM19 (Apply the knowledge acquired in advanced mathematics to the resolution of mathematical problems, as well as to physical problems with mathematical representation.) Apply the knowledge acquired in advanced mathematics to the resolution of mathematical problems, as well as to physical problems with mathematical representation.

PROGRAM

1. Hilbert spaces

1.1 Pre-Hilbert spaces.

2.2 Hilbert spaces.

2. Operators.

2.1 Linear operators.

2.2 Eigenvalues and eigenvectors.

3. Distributions

4. Introduction to group theory

4.1 Definition and motivation (symmetires)

4.2 Exemples: SO(3), SU(2), SU(N) (relation with unitary operators).

4.3 Lie algebras (generators of the continuous group)

4.4 su(N) (relation with selfadjoint operators) and relation with su(2) with so(3)

5. Representations

6. Tensorial methods

Learning activities and methodology

Title	Hours	ECTS	Learning outcomes
Practical lectures: The instructor will solve a set of select problems from a given collection. The rest will be solved by the students.	28	1.12	CM23, KM22, KM23, KM24, KM25, SM19
Blackboard lectures: the instructor will expound basic concepts and arguments for each subject, with the support of detailed examples.	56	2.24	CM23, KM22, KM23, KM24, KM25, SM19
Study of theoretical foundations	26.5	1.06	CM23, KM22, KM23, KM24, KM25, SM19
Select homework	16	0.64	CM23, KM22, KM23, KM24, KM25, SM19
Individual and group problem solving	16	0.64	CM23, KM22, KM23, KM24, KM25, SM19

This course develops mathematical language and calculation tools that are basicfor advanced physics subjects. The personal work of the student is fundamental to attaining the pertinent knowledge and skills.Classroom sessions will be divided into:Lectures: The teacher will present the basic concepts and reasoning of eachSubject, with the support of examples.Problem classes: Among a collection of problems, the teacherwill solve in detail a selection. Students will have to work on their own the rest.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title	Weight	Hours	ECTS	Learning outcomes
Final Exam	50%	2.5	0.1	CM23, KM22, KM23, KM24, KM25, SM19
Mid-Term Exam	40%	2	0.08	CM23, KM22, KM23, KM24, KM25, SM19
Homework	10%	0.5	0.02	CM23, KM22, KM23, KM24, KM25, SM19
Make-up exam	90%	2.5	0.1	CM23, KM22, KM23, KM24, KM25, SM19

Grading (Ordinary)

A) Mid-Term exam (40% of the grade): written exam, without books, individual, about mid semester.

B) Final Exam (50 % of the grade): written exam, without books, individual, at the end of the semester.

C) Take-home exercises (10% of the grade): There will be several exercises to be handed in during the semester.

The final grade will be the result of A+B+C.

D) Make-up exam (90% of the grade): If the grade achieved in A+B >3.5/10, the student will have the right to take a final make-up exam provided he/she has already taken the exams A+B. The grade achieved in this exam will replace the grade achieved in the exams A+B in all cases.

Grading (\"Avaluacio Unica\")

A) Final exam (50 % of the grade): written exam, without books, individual, at the end of the semester.

B) Oral exam (50 % of the grade) : individual exam, at the end of the semester.

C) Final oral make-up exam (100 % of the grade): oral exam, optional, at the end of the semester. If the grade achieved in A+B >3.5/10, the student will have the right to take a final make-up exam provided he/she has already taken the exams A+B. The grade achieved in this exam will replace the grade achieved in the exams A+B in all cases.

Both gradings (ordinary and \"avaluacio unica\") will have the final exam on the same day. Idem for the make-up exams.

The bonus points ("palotes") will not be added to the result of the make-up exam.

Bibliography

Basic bibliografy.

P. Szekeres, A course in Modern Mathematical Physics.

Elvira Romera et al., Métodos matemáticos: Problemas de espacios de Hilbert, operadores lineales y espectros

G. Arfken, Mathematical Methods for Physics.

Advanced and complementary bibliography.

J.J. Sakurai, Modern Quantum Mechanics.

J.F. Cornwell, Group theory in Physics.

H. Georgi, Lie Algebras in particle physics.

L. Abellanas i A. Galindo, Espais de Hilbert.

S.K. Barbarian, Introducció a l'espai de Hilbert.

L. Schwartz, Métodos Matemáticos para las ciencias físicas.

Software

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching	Group	Language	Semester	Shift
(TE) Theory	1	English	second semester	morning-mixed
(PAUL) Classroom practices	1	English	second semester	morning-mixed
(TE) Theory	2	English	second semester	morning-mixed
(PAUL) Classroom practices	2	English	second semester	morning-mixed
(SEM) Seminars	11	English	second semester	morning-mixed
(SEM) Seminars	12	English	second semester	morning-mixed
(SEM) Seminars	21	English	second semester	morning-mixed
(SEM) Seminars	22	English	second semester	morning-mixed