# MATHEMATICAL METHODS FOR PHYSICS

## Learning outcomes of the course unit

Students are supposed to get to master the mathematical skills which are essential for the study of the most advanced subjects in Physics, in particular Quantum Physics. They should understand the power of mathematical tools for tackling a variety of problems in different fields: what they learn should put them in a position to solve problems in many different circumstances. Students will be involved in solving problems in front of their colleagues during the lessons; they will be also asked to present solutions to problems they will be assigned for being worked out at home. All this is intended as a training of their communication skills (they should be able to argue in public).

## Prerequisites

Basic notions of real analysis, calculus, geometry and algebra

## Course contents summary

The contents represent a fair finalization of the mathematical training of a physicist. The program is quite broad, but it is supposed to combine mathematical rigor and fluency in organizing an overall framework (with an emphasis on computational tools).

Classical real and complex analysis will be completed with the theory of analytic functions (residues, power series, integration in the complex plane).

Main part of the course is devoted to the theory of linear operators in finite dimensional spaces (aiming at a sufficiently rigorous knowledge of the spectral theory), with the due insight in algebra and metric topology. The extension to functional spaces L1 and L2 will go through approximation problems, orthogonal functions, Fourier series and transform, trying to put contents in the perspective of a first approach to quantum mechanics.

In this frame, there will be a short course on differential equations in the complex field, possibly with applications to the Schrödinger equation. The spectral problem in Hilbert spaces will be approached without aiming at completeness.

## Course contents

Numerical Fields.

Complex Analysis. Analytic functions, basic instruments.

Residues, power series, definite integrals.

Linear manifolds, abstract vector spaces. Linear dependence. Dimension.

Real and compolex spaces. Isomorphism.

Scalar product. Orthogonality.

Metric spaces. Basic notions in topology.

Basis, orthogonal systems, orthogonalization.

Basis transformation.

Linear functional and Riesz Theorem Dirac formalism.

Sequences and convergence.

Linear applications and matrices.

Abstract linear operators.

Diagonalization.

Adjoint operators.

Eigenvalues and eigenvectors.

Hermitian, unitary and normal operators.

Projectors. Function of operators.

Complete sets of hermitian operators.

Polynomials and orthogonal functions.

Approximation. L1 and L2 spaces.

Fourier series and transforms.

Differential equations in the complex field.

Special functions.

Applications to the Schrödinger equation.

## Recommended readings

There are many excellent books on the subjects covered. A (partial) list includes

V. Smirnov, Corso di Matematica superiore, vol.III,2 (MIR)

E. Onofri, Teoria degli Operatori lineari, http://www.fis.unipr.it/home/enrico.onofri/#Lezioni

F.G.Tricomi, Metodi Matematici della Fisica (Cedam)

M.Spiegel, Variabili Complesse (Schaum, Etas)

E.Kolmogorov, S.Fomin, Elementi di teoria delle funzioni e dell'analisi funzionale (ER)

## Teaching methods

Lessons and exercises in the classroom (with students involved in working out the solutions). Extra exercises will be assigned for being worked out at home. The latter will be mostly assigned in the COMPITO mode on ELLY.

## Assessment methods and criteria

There will be an intermediate written test in the winter exams session.

Final written and oral tests.

The written test consists in exercises aiming to check the skill in calculus: they will be variations of exercises worked out during the exercise sessions.

The oral test consists in the discussion of fundamental topics, in order to prove the methodological and conceptual mastership of the latter. The oral exam will be based on the discussion of two topics: one will be assigned a few days in advance by the lecturer (mainly related to the written test results) and one chosen by the student.