# FISICA MATEMATICA SUPERIORE

**Academic Year 2019/2020**- 2° Year - Curriculum APPLICATIVO

**Teaching Staff:**

**Vittorio ROMANO**and

**Orazio MUSCATO**

**Credit Value:**6

**Scientific field:**MAT/07 - Mathematical physics

**Taught classes:**35 hours

**Exercise:**12 hours

**Term / Semester:**2°

## Learning Objectives

The aim of the course is to give the student a backgrond on advanced topics of mathematical physics concerning kinetc theory and quantum mechanics. One allows the students to complete the education in mathematical physics both in view of future research activities, for example in a PhD program, and in view of a teaching activity.

In particular, the course aims to allow the student to acquire the following skills:

**knowledge and understanding: **knowledge of results and fundamental methods in kinetic theory and quantum mechanics. Skill of understanding problems and to extract the major feature. Skill of reading, undertanding and analyzing a subject in the mathematical physics literature and present it in a clear and accurate way.

**applying knowledge and understanding:** skill of elaborating new example or solving novel theoretical exsercise, looking for the most appopriate methods and applying them in a suitable way.

**making judgements:** To be able of devise proposals suited to correctly interprete complex problems in the framework of kinetic theory and quantum mechanics and their applications. To be able to formulate autonomously adequate judgements on the applicablity of mathematical physics models to theoretical or real situations.

**communication skills**: skills of presenting arguments, problems, ideas and solutions in mathematical terms with clarity and accuracy and with procedures suited for the audience, both in an oral and a written form. Skill of clearly motivating the choice of the strategy, method and contents, along with the employed computational tools.

**learning skills: **reading and analyzing a subject in the mahematical physics literature. To tackle in an autonomuous way the systematic study of arguments not previously treated. To acquire a degree of autonomy such that the student can be able to start with an autonomuos reserach activity.

## Course Structure

The course is held through frontal lessons. Exsercises in classroom are foreseen.

Attending the lectures is strongly suggested.

Final exam: drawing up a report on a subject related to those of the corse, followed by an oral exam.

## Detailed Course Content

Part A: elements of kinetic theory

Microscopic description of an ensamble of N particles. Phase-space. Gibbs ensamble. Liouville theorem. Reduction of the Liouville equation and hints on the BBGKY hierarchy. Molecular chaos hypothesis and the Boltzamnn equation. Collisional invariants. H theorem and the Maxwellian distribution. Moments of the Boltzmann equation: conservation laws of maroscopic variables, closure relations. Direct simulation Monte carlo (hints).

- Part B: elements of quantum mechanics

Recalls of distributions, Fourier transform and Hamiltonian mechanics. States in quantum mechanics. Heisenberg uncertainty principle. Observables. Equations of Schrödinger and Heisenberg. Canonical commutation relations. Coordinates and momentum representations. Classical limit and Weyl's quantization. Examples of solutions of the Schrödinger equation: wells and hole barriers, quantum harmonic oscillator. Hints on the perturbation theory.

## Textbook Information

[1] Carlo Cercignani

Mathematical Methods in Kinetic Theory

Plenum Press

[2] L.D. Faddeev, O. A. Yakubovskii

Lectures on Quantum Mechanics for Mathematics Students

AMS