Fisica | MATHEMATICAL METHODS OF PHYSICS

cod. 0522600017

MATHEMATICAL METHODS OF PHYSICS

	0522600017
	DIPARTIMENTO DI FISICA "E.R. CAIANIELLO"
	EQF7
	PHYSICS
	2016/2017

PERCORSO COMUNE

	OBBLIGATORIO
	YEAR OF COURSE 1
	YEAR OF DIDACTIC SYSTEM 2014
	PRIMO SEMESTRE

TEACHING UNITS

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

	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 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.

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 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.

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 AND PRE-HILBERTIAN SPACES, SCHWARTZ AND TRIANGLE INEQUALITIES, COMPLETENESS AND HILBERT SPACES, SUBSPACES AND SUBMANIFOLDS, DEFINITION OF ORTHOGONALITY, PROJECTION OF A VECTOR ON A SUBSET OF A SUBSPACE AND, GRAM-SCHMIDT ORTHONORMALIZATION PROCEDURE. BESSEL'S INEQUALITY, PARSEVAL'S EQUALITY (COMPLETENESS RELATION), ORTHONORMAL BASES, COMPLETE ORTHONORMAL SYSTEMS AND EQUIVALENCE WITH RESPECT TO COMPLETENESS, DIMENSIONALITY (CARDINALITY) OF A HILBERT SPACE, SEPARABLE SPACES, COMPLETENESS OF THE SPACE L^2. OPERATORS AND LINEAR FUNCTIONALS IN HILBERT SPACES: DEFINITIONS, NORMS, AND BOUNDEDNESS. DOMAIN AND RANGE, NATURAL DOMAIN, EQUIVALENCE BETWEEN BOUNDEDNESS AND CONTINUITY, EXTENSION AND REDUCTION OF AN OPERATOR, SUMS AND PRODUCTS OF OPERATORS. LINEAR FUNCTIONALS, DUAL SPACE, AND RIESZ REPRESENTATION THEOREM. MATRIX AND INTEGRAL REPRESENTATIONS OF AN OPERATOR. COMMUTATORS AND ANTICOMMUTATORS. NECESSARY AND SUFFICIENT CONDITIONS FOR DIAGONALIZABILITY: NORMAL OPERATORS. SPECIAL INSTANCES OF NORMAL OPERATORS: UNITARY AND HERMITIAN OPERATORS. ADJOINT OF AN OPERATOR AND SELF-ADJOINT OPERATORS (BOUNDED AND UNBOUNDED). SELF-ADJOINTNESS AND HERMITIANITY: SELF-ADJOINT EXTENSIONS. ADJOINT OF A PRODUCT OF OPERATORS, PRODUCTS AND SUMS OF HERMITIAN (SELF-ADJOINT) OPERATORS, ANTI-HERMITIAN (ANTI-SELF-ADJOINT) OPERATORS, COMMUTATORS AND ANTICOMMUTATORS OF HERMITIAN (SELF-ADJOINT) OPERATORS. PROJECTORS. EIGENVALUES AND EIGENVECTORS, EIGENMANIFOLDS AND EIGENSPACES, PROPERTIES OF EIGENVALUES AND EIGENVECTORS OF HERMITIAN AND UNITARY OPERATORS. SPECTRAL THEORY: POINT SPECTRUM, CONTINUOUS SPECTRUM, RESIDUAL SPECTRUM. PROPERTIES OF THE SPECTRUM OF A SELF-ADJOINT OPERATOR. THE SPECTRUM OF THE POSITION (MOMENTUM) OPERATOR IN L^2, IMPROPER EIGENVALUES. AN EXAMPLE OF A NON-DISCRETE POINT SPECTRUM: THE ANNIHILATION OPERATOR. THE SPECTRAL THEOREM IN FINITE DIMENSIONS, SPECTRAL REPRESENTATION OF A FINITE-DIMENSIONAL HERMITIAN OPERATOR AND ITS INDUCTION IN INFINITE DIMENSIONS, SELF-ADJOINT HAMILTONIAN AS A CONDITION FOR UNITARY TIME EVOLUTIONS. POLAR AND SINGULAR VALUE REPRESENTATIONS OF AN OPERATOR. DISTRIBUTIONS: GENERAL DEFINITION OF A DISTRIBUTION THROUGH ITS REPRESENTATION, TEST FUNCTIONS AND THE SPACE OF TEST FUNCTIONS. SUMS OF DISTRIBUTIONS, THE PRODUCT OF A DISTRIBUTION AND A FUNCTION, DERIVATIVE OF A DISTRIBUTION. REPRESENTATION OF THE DIRAC DELTA, EXAMPLES OF REPRESENTATIONS. FOURIER TRANSFORM OF THE DIRAC DELTA, DERIVATIVES OF THE DIRAC DELTA. HEAVISIDE THETA-FUNCTION DEFINED AS THE DISTRIBUTION AND ITS RELATIONSHIP WITH THE DIRAC DELTA.

Contents

HILBERT SPACES:
INNER PRODUCT AND PRE-HILBERTIAN SPACES, SCHWARTZ AND TRIANGLE INEQUALITIES, COMPLETENESS AND HILBERT SPACES, SUBSPACES AND SUBMANIFOLDS, DEFINITION OF ORTHOGONALITY, PROJECTION OF A VECTOR ON A SUBSET OF A SUBSPACE AND, GRAM-SCHMIDT ORTHONORMALIZATION PROCEDURE. BESSEL'S INEQUALITY, PARSEVAL'S EQUALITY (COMPLETENESS RELATION), ORTHONORMAL BASES, COMPLETE ORTHONORMAL SYSTEMS AND EQUIVALENCE WITH RESPECT TO COMPLETENESS, DIMENSIONALITY (CARDINALITY) OF A HILBERT SPACE, SEPARABLE SPACES, COMPLETENESS OF THE SPACE L^2.

OPERATORS AND LINEAR FUNCTIONALS IN HILBERT SPACES:
DEFINITIONS, NORMS, AND BOUNDEDNESS. DOMAIN AND RANGE, NATURAL DOMAIN, EQUIVALENCE BETWEEN BOUNDEDNESS AND CONTINUITY, EXTENSION AND REDUCTION OF AN OPERATOR, SUMS AND PRODUCTS OF OPERATORS. LINEAR FUNCTIONALS, DUAL SPACE, AND RIESZ REPRESENTATION THEOREM. MATRIX AND INTEGRAL REPRESENTATIONS OF AN OPERATOR. COMMUTATORS AND ANTICOMMUTATORS. NECESSARY AND SUFFICIENT CONDITIONS FOR DIAGONALIZABILITY: NORMAL OPERATORS. SPECIAL INSTANCES OF NORMAL OPERATORS: UNITARY AND HERMITIAN OPERATORS. ADJOINT OF AN OPERATOR AND SELF-ADJOINT OPERATORS (BOUNDED AND UNBOUNDED). SELF-ADJOINTNESS AND HERMITIANITY: SELF-ADJOINT EXTENSIONS. ADJOINT OF A PRODUCT OF OPERATORS, PRODUCTS AND SUMS OF HERMITIAN (SELF-ADJOINT) OPERATORS, ANTI-HERMITIAN (ANTI-SELF-ADJOINT) OPERATORS, COMMUTATORS AND ANTICOMMUTATORS OF HERMITIAN (SELF-ADJOINT) OPERATORS. PROJECTORS. EIGENVALUES AND EIGENVECTORS, EIGENMANIFOLDS AND EIGENSPACES, PROPERTIES OF EIGENVALUES AND EIGENVECTORS OF HERMITIAN AND UNITARY OPERATORS. SPECTRAL THEORY: POINT SPECTRUM, CONTINUOUS SPECTRUM, RESIDUAL SPECTRUM. PROPERTIES OF THE SPECTRUM OF A SELF-ADJOINT OPERATOR. THE SPECTRUM OF THE POSITION (MOMENTUM) OPERATOR IN L^2, IMPROPER EIGENVALUES. AN EXAMPLE OF A NON-DISCRETE POINT SPECTRUM: THE ANNIHILATION OPERATOR. THE SPECTRAL THEOREM IN FINITE DIMENSIONS, SPECTRAL REPRESENTATION OF A FINITE-DIMENSIONAL HERMITIAN OPERATOR AND ITS INDUCTION IN INFINITE DIMENSIONS, SELF-ADJOINT HAMILTONIAN AS A CONDITION FOR UNITARY TIME EVOLUTIONS. POLAR AND SINGULAR VALUE REPRESENTATIONS OF AN OPERATOR.

DISTRIBUTIONS:
GENERAL DEFINITION OF A DISTRIBUTION THROUGH ITS REPRESENTATION, TEST FUNCTIONS AND THE SPACE OF TEST FUNCTIONS. SUMS OF DISTRIBUTIONS, THE PRODUCT OF A DISTRIBUTION AND A FUNCTION, DERIVATIVE OF A DISTRIBUTION. REPRESENTATION OF THE DIRAC DELTA, EXAMPLES OF REPRESENTATIONS. FOURIER TRANSFORM OF THE DIRAC DELTA, DERIVATIVES OF THE DIRAC DELTA. HEAVISIDE THETA-FUNCTION DEFINED AS THE DISTRIBUTION AND ITS RELATIONSHIP WITH THE DIRAC DELTA.

	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
	N.I. AKHIEZER AND I.M. GLAZMAN: "THEORY OF LINEAR OPERATORS IN HILBERT SPACE", DOVER PUBLICATIONS. R. ALICKI AND M. FANNES: "QUANTUM DYNAMICAL SYSTEMS", OXFORD UNIVERSITY PRESS. I. BENGTSSON AND K. ZYCZKOWSKI: "GEOMETRY OF QUANTUM STATES", CAMBRIDGE UNIVERSITY PRESS. 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. M.A. NIELSEN AND I.L. CHUANG: "QUANTUM COMPUTATION AND QUANTUM INFORMATION", CAMBRIDGE UNIVERSITY PRESS. F. RIESZ AND B.S. NAGY: "FUNCTIONAL ANALYSIS", DOVER PUBLICATIONS. C. ROSSETTI: "METODI MATEMATICI DELLA FISICA", LIBRERIA EDITRICE UNIVERSITARIA LEVROTTO & BELLA. W. RUDIN: "REAL AND COMPLEX ANALYSIS", MC GRAW-HILL.

Texts

N.I. AKHIEZER AND I.M. GLAZMAN: "THEORY OF LINEAR OPERATORS IN HILBERT SPACE", DOVER PUBLICATIONS.
R. ALICKI AND M. FANNES: "QUANTUM DYNAMICAL SYSTEMS", OXFORD UNIVERSITY PRESS.
I. BENGTSSON AND K. ZYCZKOWSKI: "GEOMETRY OF QUANTUM STATES", CAMBRIDGE UNIVERSITY PRESS.
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.
M.A. NIELSEN AND I.L. CHUANG: "QUANTUM COMPUTATION AND QUANTUM INFORMATION", CAMBRIDGE UNIVERSITY PRESS.
F. RIESZ AND B.S. NAGY: "FUNCTIONAL ANALYSIS", DOVER PUBLICATIONS.
C. ROSSETTI: "METODI MATEMATICI DELLA FISICA", LIBRERIA EDITRICE UNIVERSITARIA LEVROTTO & BELLA.
W. RUDIN: "REAL AND COMPLEX ANALYSIS", MC GRAW-HILL.