Fisica | FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

cod. 0512600031

FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

	0512600031
	DIPARTIMENTO DI FISICA "E.R. CAIANIELLO"
	EQF6
	PHYSICS
	2017/2018

PERCORSO COMUNE Cohort 2015/2016

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2010
	ANNUALE

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
FIS/02	7	56	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
FIS/02	2	24	EXERCISES	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

	Objectives
	THE COURSE INTENDS TO TRANSFER THE MATHEMATICAL KNOWLEDGE THAT IS NECESSARY IN THE COMPREHENSION OF PHYSICAL PHENOMENA OF HIGHER COMPLEXITY WITH RESPECT TO THE ONES FACED IN THE FIRST TWO YEARS. KNOLEDGE AND COMPREHENSION ABILITY: THE AIM OF THE COURSE IS TO RENDER THE STUDENT ABLE TO APPLY THE KNOWLEDGE OF MATHEMATICAL TYPE AND METHODS FOR THE COMPREHENSION, AT AN ADVANCED LEVEL, OF SOME ASPECTS OF QUANTUM PHYSICS AND FOR THE SOLUTION OF EXCERSISES AND PROBLEMS IN THIS FIELDS. THE COURSE AIMS TO GIVE THE STUDENTS THE MATHEMATICAL COMPETENCES USEFUL IN A WORKING FRAMEWORK. APPLYING KNOWLEDGE AND COMPREHENSION: THE STUDENT WILL BE ABLE TO APPLY HIS/HER KNOWLEDGE TO SOLVE ADVANCED PHYSICAL PROBLEMS THAT IMPLY THE USE OF FOURIER SERIES, LAPLACE AND FOURIER TRANSFORM, DIFFERENTIAL EQUATIONS AT PARTIAL DERIVATIVES, INTEGRALS IN THE COMPLEX PLANE, BOTH IN THE STUDY COURSE AND IN A WORKING ENVIRONMENT.

THE COURSE INTENDS TO TRANSFER THE MATHEMATICAL KNOWLEDGE THAT IS NECESSARY IN THE COMPREHENSION OF PHYSICAL PHENOMENA OF HIGHER COMPLEXITY WITH RESPECT TO THE ONES FACED IN THE FIRST TWO YEARS.

KNOLEDGE AND COMPREHENSION ABILITY:
THE AIM OF THE COURSE IS TO RENDER THE STUDENT ABLE TO APPLY THE KNOWLEDGE OF MATHEMATICAL TYPE AND METHODS FOR THE COMPREHENSION, AT AN ADVANCED LEVEL, OF SOME ASPECTS OF QUANTUM PHYSICS AND FOR THE SOLUTION OF EXCERSISES AND PROBLEMS IN THIS FIELDS. THE COURSE AIMS TO GIVE THE STUDENTS THE MATHEMATICAL COMPETENCES USEFUL IN A WORKING FRAMEWORK.

APPLYING KNOWLEDGE AND COMPREHENSION:
THE STUDENT WILL BE ABLE TO APPLY HIS/HER KNOWLEDGE TO SOLVE ADVANCED PHYSICAL PROBLEMS THAT IMPLY THE USE OF FOURIER SERIES, LAPLACE AND FOURIER TRANSFORM, DIFFERENTIAL EQUATIONS AT PARTIAL DERIVATIVES, INTEGRALS IN THE COMPLEX PLANE, BOTH IN THE STUDY COURSE AND IN A WORKING ENVIRONMENT.

	Prerequisites
	MATHEMATICAL COURSES LIKE ANALYSIS I AND II, GEOMETRY, AND PHYSICS COURSES: QUANTUM MECHANICS. ARGUMENTS: COMPLEX AND REAL NUMBERS, INTEGRAL AND DIFFERENTIAL CALCULUS (AT ONE OR MORE VARIABLES), STUDY OF A FUNCTION, LINEAR ALGEBRA . BASIC KNOWLEDGE OF QUANTUM MECHANICS: ASSIOMS, SCHROEDINGER EQUATION, DYNAMIC VARIABLES.

	Contents
	METRIC AND TOPOLOGICAL SPACES, BANACH SPACES. INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRALS. COMPLEX ANALYSIS: ANALYTIC FUNCTIONS, CAUCHY-RIEMAN EQUATIONS, PATH INTEGRALS, DOMAINS AND CONTOURS, CAUCHY THEOREM, LIOUVILLE THEOREM. THE CONFORMAL TRANSFORMATIONS. ISOLATED SINGULARITY AND POLES. RESIDUES AND RESIDUES THEOREM. ANALYTIC EXTENSIONS. FOURIER TRANSFORMS AND FOURIER SERIES, LAPLACE TRANSFORMS, LINEAR DIFFERENTIAL EQUATIONS. HILBERT SPACES IN TWO DIMENSIONS: HILBERT SPACE C^2. DEFINITION OF OPERATORS (HERMITEAN, UNITARY, PROJECTION). BLOCH SPHERE, PAULI MATRICES, TWO-LEVEL SYSTEMS, OBSERVABLES AND THEIR MEAN-VALUE.

Contents

METRIC AND TOPOLOGICAL SPACES, BANACH SPACES. INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRALS.

COMPLEX ANALYSIS: ANALYTIC FUNCTIONS, CAUCHY-RIEMAN EQUATIONS, PATH INTEGRALS, DOMAINS AND CONTOURS, CAUCHY THEOREM, LIOUVILLE THEOREM. THE CONFORMAL TRANSFORMATIONS. ISOLATED SINGULARITY AND POLES. RESIDUES AND RESIDUES THEOREM. ANALYTIC EXTENSIONS.

FOURIER TRANSFORMS AND FOURIER SERIES, LAPLACE TRANSFORMS, LINEAR DIFFERENTIAL EQUATIONS.

HILBERT SPACES IN TWO DIMENSIONS: HILBERT SPACE C^2. DEFINITION OF OPERATORS (HERMITEAN, UNITARY, PROJECTION). BLOCH SPHERE, PAULI MATRICES, TWO-LEVEL SYSTEMS, OBSERVABLES AND THEIR MEAN-VALUE.

	Teaching Methods
	FRONTAL LESSONS AND TRAINING

	Verification of learning
	WRITTEN EXAMINATION WITH THE RESOLUTION OF EXCERCISES AND ORAL EXAMINATION ADDRESSED TO VERIFY THE ATTIDUDE IN A LOGICAL-MATHEMATICAL PRESENTATION.

	Texts
	C. ROSSETTI: METODI MATEMATICI DELLA FISICA, LIBRERIA EDITRICE UNIVERSITARIA LEVROTTO & BELLA (TORINO). W. RUDIN: REAL AND COMPLEX ANALYSIS, MC GRAW-HILL. G. G. N. ANGILELLA: ESERCIZI DI METODI MATEMATICI DELLA FISICA, SPRINGER G. CICOGNA: METODI MATEMATICI DELLA FISICA, SPRINGER

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