Advanced mathematical methods
Code: 100167Credits: 5
| Degree programme | Type | Course |
|---|---|---|
| Physics | OP | 3 |
Contact lecturer
- Name :
- Pere Masjuan Queralt
- Email :
- pere.masjuan@uab.cat
Teaching staff
- Pere Masjuan Queralt
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
- Communicate complex information in an effective, clear and concise manner, either orally, in writing or through ICTs, in front of both specialist and general publics.
- Use critical reasoning, show analytical skills, correctly use technical language and develop logical arguments
- Determine the symmetry group (exact or approximate) associated with a physical system.
- Determine the effect on the observables of symmetry transformation.
- Identify symmetry groups associated with the laws of physics.
- Determine the representation that characterizes a particular physical system.
- Determine the observables that characterise representation.
- Relate the symmetries of nature with the appropriate symmetry group (exact or approximate).
- Identify symmetry groups in addition to their particle and crystallography representations, associated with atomic physics.
- Identify symmetry groups associated with theories of fundamental interactions.
- Classify representations of the most simple groups.
- Relate continuous groups with the Lie algebra to which they are associated.
- Obtain representation of simple symmetry groups.
- Use the tensor calculus.
- Work independently, take initiative itself, be able to organize to achieve results and to plan and execute a project.
- Identify situations in which a change or improvement is needed.
- Identify the social, economic and environmental implications of academic and professional activities within one's own area of knowledge.
- Explain the explicit or implicit code of practice of one's own area of knowledge.
Contents
In this subject we propose a paradigm shift in university learning. Instead of theory → exercises → exam, which involves memorizing how to solve solved problems, we will use research problem → need for a mathematical tool → collective construction of knowledge → formalization → return to research. This change is profound because it presents mathematics not as a set of contents that must be memorized, but as a language that is developed to answer questions in physics. This idea, in addition to being aligned with inquiry-based learning and with the practices of Peter Liljedahl's Building Thinking Classrooms, is also a very faithful way to get closer to how physicists and mathematicians really work in research.
5 doctoral students will present an open research problem that they frame in the context of the subject. In the classroom, we will try to provide ideas so that these students can continue their research, without looking for a solution but by making realistic proposals and framed in the following program. Each of the doctoral students will deal with one of the topics. The assessment will also follow the Thinking Assessment scheme.
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*
(*If there is no time, we will propose an extra class, outside of school hours and voluntary, of 2 hours)
Learning activities and methodology
| Title | Hours | ECTS | Learning outcomes |
|---|---|---|---|
| Selective homework | 11 | 0.44 | |
| Individual and groupal work solving problems | 28 | 1.12 | |
| Study of teoretical foundations | 37 | 1.48 | |
| Blackboard lectures: the profesor will expone basic concepts and arguments for each subject, with the support of detailed examples. | 7 | 0.28 | |
| Practical lectures: Among a collection of open problems, the class group solves them collectively and we draw conclusions that build on the knowledge of the subject. | 34 | 1.36 |
This subject develops mathematical language and calculation tools that are basic for advanced Physics subjects. The student's personal work is essential to achieve the relevant knowledge and skills. For this reason, in this subject the face-to-face class sessions will be based on the teaching innovation project "From the Club to the Classroom", which is based on the Thinking Classrooms method.
The "From the Club to the Classroom" project therefore proposes to transfer the typical dynamics of the Physics Club to the classroom within the Advanced Mathematical Methods subject of the old plan of the Physics Degree,
converting the course topics into open questions related to real doctoral research problems. In this way, mathematical concepts appear as tools to address
current scientific questions.
The objectives of the project are:
1. Increase the active participation of students in the classroom through collaborative work sessions based on open research questions.
2. Reduce absenteeism in the subject through a participatory seminar model that requires direct student involvement during the sessions.
3. Improve conceptual understanding of mathematical methods used in theoretical physics, connecting them to real research problems.
4. Promote critical thinking and scientific discussion through collaborative problem-solving activities.
5. Integrate doctoral students in the teaching process, generating interaction between different levels of training within the academic community.
6. Evaluate the pedagogical impact of the methodology through a comparative study between an experimental group and a control group.
The project methodology is based on organizing the sessions as collaborative work seminars inspired by the Thinking Classrooms practices described by Peter Liljedahl in
Building Thinking Classrooms in Mathematics. Each of the five topics of the subject will be presented as an open question related to the research of a doctoral student.
The sessions will begin with a brief presentation of the scientific context of the problem by the doctoral student. Subsequently, students will work in small randomly formed groups to
explore possible mathematical strategies. Ideas will be developed on visible work surfaces (whiteboards or digital support) to facilitate discussion between groups and the circulation of ideas
in the classroom.
The teacher will act as a facilitator of the process, introducing relevant mathematical concepts when necessary and connecting the ideas generated with the formal contents of the subject
(which we will have available thanks to the subject's Moodle classroom). The sessions will end with a conceptual consolidation phase and a brief presentation of the groups' results.
In this subject, the use of Artificial Intelligence (AI) technologies is allowed as an integral part of the development of the work, provided that the final result reflects a significant contribution by the student in the analysis and personal reflection. The student must clearly identify which parts have been generated with this technology, specify the tools used and include a critical reflection on how these have influenced the process and the final result of the activity. The lack of transparency in the use of AI will be considered a lack of academic honesty and may lead to a penalty in the grade of the activity, or greater sanctions in serious cases.
Assessment
Continuous assessment activities
| Title | Weight | Hours | ECTS | Learning outcomes |
|---|---|---|---|---|
| Homework | 5% | 0.25 | 0.01 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 |
| Midterm exam | 45% | 2.25 | 0.09 | 1, 2, 14, 15, 16, 17, 18 |
| Final exam | 50% | 2.5 | 0.1 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 |
| Make-up exam | 95% | 3 | 0.12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 |
The assessment of this subject attempts to emulate the creative process of research and will take into account the following five dimensions:
active and sustained participation;
quality of mathematical reasoning;
ability to work and discuss with others;
oral and written scientific communication;
ability to relate the course contents to a physics problem.
In this subject we do not seek to obtain a good grade because we "solve the problem", but because we learn to think mathematically about a research problem.
We will follow the following guidelines:
A) 20% Active participation in the classroom
It is not simply attending. The following are assessed:
participation in discussions
involvement in the group
contribution of ideas
ability to listen and build on the ideas of others
It is assessed daily. Not participating in class implies not having this grade.
B) 15% Workbook (portfolio)
Each student maintains a document during the semester.
It is not a summary of theory but a document where you write
ideas that have appeared,
conjectures,
connections,
interesting errors,
what you have learned.
It is submitted on the day of the final exam.
C) 20% Oral presentations
Each student (or pair, or triplet) presents one of the five problems.
It does not explain theory but:
what you have tried,
why,
what worked,
what did not.
D) 20% Mathematical report
At the end of each topic (or two topics) the groups submit 4-6 pages where they explain
the problem,
the ideas explored,
the mathematical framework,
the limitations,
possible future lines.
This document is what the doctoral student receives.
E) 25% Individual consolidation test
The final test of the subject will last 3 hours and will consist of an individual written test with consultation of paper notes, designed to assess the consolidation of the concepts worked on during the course and the ability to use them in an integrated manner.
Unlike a traditional exam focused on the reproduction of demonstrations or procedures, this test will pose new situations related to theoretical physics that will require identifying, justifying and combining the mathematical tools studied in the subject. The quality of reasoning, the justification of the decisions made, the ability to establish connections between different course contents and the clarity of the mathematical argumentation will be particularly valued.
The test will be structured in three parts:
Conceptual questions, aimed at checking the understanding of the fundamental concepts.
An integrative problem, in which it will be necessary to analyze a new situation and propose a reasoned mathematical strategy to address it.
A synthesis question, designed to relate different mathematical tools of the course and reflect on their role in the formulation and resolution of physics problems.
The objective of the test is to assess the ability to think mathematically when faced with a problem, rather than the memorization of results or demonstrations.
Single Assessment
A) Final Exam (45% of the final grade): it is a written exam, without books, individual, at the end of the semester.
B) Oral Exam (55% of the final grade): it is an individual exam, at the end of the semester.
C) Oral Remedial Exam (100% of the final grade): it is an oral exam, optional, at the end of the semester. If the A+B grade >3.5/10, the student may opt for this exam as long as he or she has appeared for A+B. The grade of this exam will replace the A+B grade (single assessment) in all cases.
Both assessments will have the final exam on the same day. Ditto for the retake 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
Not concieved.
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 | first semester | morning-mixed |
| (PAUL) Classroom practices | 1 | English | first semester | morning-mixed |