MATHEMATICAL METHODS FOR PHYSICS: ELEMENTS OF FUNCTIONAL ANALYSIS WITH EXERCISE SESSIONS

Degree course: 
Corso di First cycle degree in Physics
Academic year when starting the degree: 
2022/2023
Year: 
2
Academic year in which the course will be held: 
2023/2024
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Basic elements of linear algebra, and of the theory of finite dimensional vector spaces.

Final Examination: 
Orale

The exam will consist of a written test made up of 4 exercises to be performed in 3 hours and an oral test. At least 6 tests will be organized per academic year. The positive outcome of the written test is preliminary to the oral test.

As part of the written test, within each exercise it may also be required to provide some notion of theory or some theoretical demonstration. Each exercise will consist of several questions. In the written test, each exercise will be assigned a score (expressed in out of thirty) and explicitly indicated on the exam topic. The sum of the points assigned to each exercise will be 30/30. For the purposes of the evaluation of the written test, the following criteria will be considered, in order of priority:
1) the correctness and explanation of the procedures used to solve the problems;
2) the reliability of the results obtained;
3) the correctness of the calculations and of the final result in carrying out the same;
4) the correct use of technical terminology.
The written test will be considered positive with a mark higher than or equal to 16/30.

During the oral exam, the student will be questioned about the program. First, it will be asked to comment on any errors made in the written test. Then 4 or 5 questions relating to the content of the course will be asked to the student. In particular, some demonstrations and some calculations will be asked. The final grade will be assigned by making a weighted average of the assessments obtained in the written and oral tests. The weight assigned to the written test is 1/3, while that given to the oral exam is 2/3. The exam will be considered passed if the result of this weighted average is at least 18/30. To obtain honors, the student must produce a perfect written test and be able, during the oral exam, to answer questions with a high level of difficulty and to carry out non-trivial calculations.

Assessment: 
Voto Finale

The students of this course are expected to acquire a certain operational familiarity
with those mathematical tools which, although commonly used in quantum mechanics, are nevertheless advanced enough to exceed the limits of basic courses in mathematical analysis; nevertheless maintaining a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use of it, beyond the pure and simple formal manipulation.

The course consists of a synthetic but logically coherent introduction to the elements of Functional Analysis on which the mathematical formalism of quantum mechanics is based. The traditional elementary theory of Hilbert spaces will be accompanied by an "effective" introduction to the theory of temperate distributions. Without renouncing logical coherence, the use of abstract aspects of the theory of topological vector spaces will be minimized, while the emphasis will be placed on the most frequently used elementary distributional techniques, such as the Fourier transform, and Green functions.

- Linear algebra;
- Theory of measure;
- Hilbert spaces;
- Fourier transform;
- Distribution theory.

The lessons will be in presence, with presentation of the contents also with detailed execution of the calculations. There will be no lesson hours dedicated to exercises, but exercises will be carried out by the teacher and by the students during the course.

Written notes are provided by the teacher during the course.