# Fisica | MATHEMATICAL METHODS OF PHYSICS

## Fisica MATHEMATICAL METHODS OF PHYSICS

 0522600017 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" EQF7 PHYSICS 2017/2018

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2017 PRIMO SEMESTRE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY FIS/02 5 40 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS FIS/02 1 12 EXERCISES COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 MASSIMO BLASONE T
Objectives
KNOWLEDGE AND UNDERSTANDING:
THE COURSE AIMS TO PROVIDE ADVANCED MATHEMATICAL KNOWLEDGE RELATED TO ALGEBRAS OF OBSERVABLES, HILBERT SPACES, LINEAR OPERATORS IN HILBERT SPACES AND THEORY OF DISTRIBUTIONS. SOME BASIC ELEMENTS OF QUANTUM INFORMATION THEORY AND AN INTRODUCTION TO GROUP THEORY ARE ALSO GIVEN.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE COURSE AIMS TO MAKE STUDENTS ABLE TO USE THE ACQUIRED KNOWLEDGE AND METHODS FOR THE UNDERSTANDING OF QUANTUM PHYSICS AT AN ADVANCED LEVEL, AND FOR THE SOLUTION OF EXERCISES AND PROBLEMS IN THIS FIELD.
Prerequisites
MATHEMATICAL COURSES FROM THE BACHELOR DEGREE. IN PARTICULAR: CALCULUS I, II, III, IV, GEOMETRY I AND II, MATHEMATICAL METHODS OF PHYSICS (BACHELOR LEVEL). THIRD-YEAR COURSES IN PHYSICS: QUANTUM PHYSICS.
TOPICS: REAL AND COMPLEX NUMBERS, DIFFERENTIAL AND INTEGRAL CALCULUS (SINGLE AND MULTIPLE VARIABLES), THE STUDY OF FUNCTIONS, SEQUENCES AND SERIES (NUMERIC AND FUNCTIONS), LINEAR ALGEBRA AND LINEAR SPACES, ANALYTIC GEOMETRY, COMPLEX PLANE, TRANSFORM, AND DIFFERENTIAL EQUATIONS. ELEMENTARY KNOWLEDGE OF QUANTUM MECHANICS: AXIOMS, SCHROEDINGER EQUATION, OBSERVABLES: POSITION, MOMENTUM, ANGULAR MOMENTUM, AND SPIN. SOLUTION OF SIMPLE PROBLEMS IN EXTERNAL POTENTIAL.
Contents
HILBERT SPACES:
INNER PRODUCT, SCHWARTZ AND TRIANGLE INEQUALITIES, COMPLETENESS. ORTHONORMAL BASES. COMPLETE ORTHONORMAL SYSTEMS. SEPARABLE SPACES.

OPERATORS AND LINEAR FUNCTIONALS IN HILBERT SPACES: DEFINITIONS, NORMS, AND BOUNDEDNESS. DOMAIN AND RANGE. LINEAR FUNCTIONALS, DUAL SPACE, AND RIESZ REPRESENTATION THEOREM. UNITARY AND HERMITIAN OPERATORS. ADJOINT OF AN OPERATOR AND SELF-ADJOINT OPERATORS. PROJECTORS. SPECTRAL THEORY: POINT SPECTRUM, CONTINUOUS SPECTRUM, RESIDUAL SPECTRUM. PROPERTIES OF THE SPECTRUM OF A SELF-ADJOINT OPERATOR.

DISTRIBUTIONS:
GENERAL DEFINITION, TEST FUNCTIONS AND THE SPACE OF TEST FUNCTIONS. DIRAC DELTA

QUANTUM INFORMATION: ENTANGLED STATES. QUANTIFICATION OF ENTANGLEMENT. MULTIPARTITE ENTANGLEMENT. GEOMETRY OF QUANTUM STATES.

ELEMENTS OF GROUP THEORY: DEFINITIONS. REPRESENTATIONS. FINITE GROUPS. LIE GROUPS. LIE ALGEBRAS AND THEIR REPRESENTATIONS.
Teaching Methods
FRONT LESSON CLASSES AND EXERCISE CLASSES. THE STUDENT IS OBLIGED TO FOLLOW THE LECTURES AND TO PARTICIPATE WITH HIS CONTRIBUTIONS TO THE DISCUSSIONS DURING THE PRESENTATIONS OF SPECIALISTIC SUBJECTS. THE PERMANENT INTERACTION WITH THE STUDENT AND THE COMMON DISCUSSIONS DURING THE LECTURES ALLOWS A NON SUPERFICIAL JUDGEMENT OF THE STUDENT PREPARATION IN ITINERE
Verification of learning
THE FINAL ORAL EXAMINATION INCLUDES THE RESOLUTION OF SOME PROBLEMS ON HILBERT SPACES AND LINEAR OPERATORS. THE STUDENT IS REQUIRED TO BE ABLE TO EXPOSE IN A CLEAR AND SYNTETIC WAY THE ARGUMENTS PRESENTED IN THE LECTURES AND TO FORMULATE AUTONOMOUS JUDGEMENTS.
Texts
S. DE SIENA, AN INTRODUCTION TO HILBERT SPACES (WITH EXERCISES AND COMPLEMENTS) - NOTE
G. CICOGNA, "METODI MATEMATICI DELLA FISICA", SPRINGER-VERLAG
C. ROSSETTI: "METODI MATEMATICI DELLA FISICA", LIBRERIA EDITRICE UNIVERSITARIA LEVROTTO & BELLA.
M.A. NIELSEN AND I.L. CHUANG: "QUANTUM COMPUTATION AND QUANTUM INFORMATION", CAMBRIDGE UNIVERSITY PRESS.

N.I. AKHIEZER AND I.M. GLAZMAN: "THEORY OF LINEAR OPERATORS IN HILBERT SPACE", DOVER PUBLICATIONS.
R. COURANT AND D. HILBERT: "METHODS OF MATHEMATICAL PHYSICS", VOLUMES 1 & 2, WILEY-VCH PUBLISHERS.
L. MACCONE E L. SALASNICH: "MECCANICA QUANTISTICA, CAOS E SISTEMI COMPLESSI", CAROCCI EDITORE.
V. MORETTI: "TEORIA SPETTRALE E MECCANICA QUANTISTICA", SPRINGER ITALIA.
F. RIESZ AND B.S. NAGY: "FUNCTIONAL ANALYSIS", DOVER PUBLICATIONS.
W. RUDIN: "REAL AND COMPLEX ANALYSIS", MC GRAW-HILL.