Fisica | FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

cod. 0512600031

FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

	0512600031
	DIPARTIMENTO DI FISICA "E.R. CAIANIELLO"
	EQF6
	PHYSICS
	2018/2019

PERCORSO COMUNE Cohort 2016/2017

	OBBLIGATORIO
	YEAR OF COURSE 3
	YEAR OF DIDACTIC SYSTEM 2016
	PRIMO SEMESTRE

TEACHING UNITS

	SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
	FIS/02	9	72	LESSONS	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, CONFORMAL TRANSFORMATION BOTH IN THE STUDY COURSE AND IN A WORKING ENVIRONMENT. JUDGMENT AUTONOMY KNOWING HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE ADVANCED PHYSICS PROBLEMS IN THE FIELD OF QUANTUM MECHANICS AND IN APPLIED FIELDS SUCH AS FLUID DYNAMICS, ELECTRODYNAMICS AND ACOUSTICS. COMMUNICATION SKILLS TO BE ABLE TO SOLVE WRITTEN EXERCISES, IN A SYNTHETIC WAY, AND TO EXPOSE ORALLY WITH PROPERTIES OF LANGUAGE THE OBJECTIVES, THE PROCEDURE AND THE RESULTS OF THE ELABORATIONS CARRIED OUT. ABILITY TO LEARN BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE COURSE, AND DEEPEN THE TOPICS COVERED USING MATERIALS OTHER THAN THOSE PROPOSED.

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, CONFORMAL TRANSFORMATION BOTH IN THE STUDY COURSE AND IN A WORKING ENVIRONMENT.

JUDGMENT AUTONOMY
KNOWING HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE ADVANCED PHYSICS PROBLEMS IN THE FIELD OF QUANTUM MECHANICS AND IN APPLIED FIELDS
SUCH AS FLUID DYNAMICS, ELECTRODYNAMICS AND ACOUSTICS.

COMMUNICATION SKILLS
TO BE ABLE TO SOLVE WRITTEN EXERCISES, IN A SYNTHETIC WAY, AND TO EXPOSE ORALLY WITH PROPERTIES OF
LANGUAGE THE OBJECTIVES, THE PROCEDURE AND THE RESULTS OF THE ELABORATIONS CARRIED OUT.

ABILITY TO LEARN
BEING ABLE TO APPLY THE ACQUIRED KNOWLEDGE TO CONTEXTS DIFFERENT FROM THOSE PRESENTED DURING THE
COURSE, AND DEEPEN THE TOPICS COVERED USING MATERIALS OTHER THAN THOSE PROPOSED.

	Prerequisites
	IS REQUIRED SOME KNOWLEDGE FROM MATHEMATICAL COURSES AS ANALYSIS I AND II, GEOMETRY, AND PHYSICS COURSES AS 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
	LINEAR VECTORIAL SPACES. INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRALS; L² FUNCTIONS SPACE. [10 H; EXC. 4H] FOURIER TRANSFORMS AND FOURIER SERIES. GIBBS PHENOMENON. SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH FOURIER SERIES AND TRANSFORMS: WAVE EQUATION ON A LINE; DIFFUSION EQUATION WITHOUT AND WITH SOURCE; WAVE EQUATION WITH A DISSIPATIVE TERM AND WITH PERIODIC BOUNDARY CONDITIONS. [14H; EXC. 8H] 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. [6H] COMPLEX ANALYSIS: ANALYTIC FUNCTIONS, CAUCHY-RIEMAN EQUATIONS, PATH INTEGRALS, DOMAINS AND CONTOURS, CAUCHY THEOREM. THE CONFORMAL TRANSFORMATIONS. ISOLATED SINGULARITY AND POLES. RESIDUES AND RESIDUES THEOREM. ANALYTIC EXTENSIONS. [20H; EXC. 10H] LAPLACE TRANSFORM AND ANTITRANSFORM, LINEAR DIFFERENTIAL EQUATIONS. [4H; EXC. 2H]

Contents

LINEAR VECTORIAL SPACES. INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRALS; L² FUNCTIONS SPACE. [10 H; EXC. 4H]
FOURIER TRANSFORMS AND FOURIER SERIES. GIBBS PHENOMENON. SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH FOURIER SERIES AND TRANSFORMS: WAVE EQUATION ON A LINE; DIFFUSION EQUATION WITHOUT AND WITH SOURCE; WAVE EQUATION WITH A DISSIPATIVE TERM AND WITH PERIODIC BOUNDARY CONDITIONS. [14H; EXC. 8H]

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. [6H]

COMPLEX ANALYSIS: ANALYTIC FUNCTIONS, CAUCHY-RIEMAN EQUATIONS, PATH INTEGRALS, DOMAINS AND CONTOURS, CAUCHY THEOREM. THE CONFORMAL TRANSFORMATIONS. ISOLATED SINGULARITY AND POLES. RESIDUES AND RESIDUES THEOREM. ANALYTIC EXTENSIONS. [20H; EXC. 10H]

LAPLACE TRANSFORM AND ANTITRANSFORM, LINEAR DIFFERENTIAL EQUATIONS.
[4H; EXC. 2H]

	Teaching Methods
	FRONTAL LESSONS AND TRAINING. IN THE THEORETICAL LESSONS THE TOPICS ARE PRESENTED INTRODUCING NEW OR PROBLEMS OF GROWING COMPLEXITY. IN EXERCISES IT IS CONSIDERED A PROBLEM TO BE SOLVED USING THE PRESENTED TECHNIQUES IN THE THEORETICAL LESSONS. THE DEVELOPMENT OF THE PROBLEM IS GUIDED BY THE TEACHER AND AIMS TO INVOLVE STUDENTS TO DEVELOP AND STRENGTHEN THE STUDENT'S ABILITY TO IDENTIFY THE MOST SUITABLE TECHNIQUES FOR THE SOLUTION.

Teaching Methods

FRONTAL LESSONS AND TRAINING.

IN THE THEORETICAL LESSONS THE TOPICS ARE PRESENTED INTRODUCING NEW OR PROBLEMS OF GROWING COMPLEXITY. IN EXERCISES IT IS CONSIDERED A PROBLEM TO BE SOLVED USING THE PRESENTED TECHNIQUES
IN THE THEORETICAL LESSONS. THE DEVELOPMENT OF THE PROBLEM IS GUIDED BY THE TEACHER AND AIMS TO INVOLVE STUDENTS TO DEVELOP AND STRENGTHEN
THE STUDENT'S ABILITY TO IDENTIFY THE MOST SUITABLE TECHNIQUES FOR THE SOLUTION.

	Verification of learning
	WRITTEN EXAMINATION WITH THE RESOLUTION OF EXCERCISES AND ORAL EXAMINATION ADDRESSED TO VERIFY THE ATTIDUDE IN A LOGICAL-MATHEMATICAL PRESENTATION. THE ORAL EXAMINATION IS AIMED AT DEEPENING THE LEVEL OF THE THEORETICAL KNOWLEDGE, AUTONOMY OF ANALYSIS AND JUDGMENT, AS WELL AS THE STUDENT'S EXHIBITION SKILLS. THE LEVEL OF EVALUATION OF THE EXAMINATION TAKES INTO ACCOUNT THE EFFICIENCY OF THE METHODS USED, THE ACCURACY OF THE ANSWERS, AND CLARITY IN THE PRESENTATION. THE MINIMUM LEVEL OF ASSESSMENT (18) IS ASSIGNED WHEN THE STUDENT DEMONSTRATES UNCERTAINTIES IN THE APPLICATION OF THE METHODS FOR SOLVING THE PROPOSED EXERCISES AND HAS A LIMITED KNOWLEDGE OF THE MAIN THEOREMS UNDERLYING THE APPLICATIONS. THE MAXIMUM LEVEL (30) IS ATTRIBUTED WHEN THE STUDENT SHOWS A COMPLETE AND DEEP KNOWLEDGE OF THE RESOLUTION OF INTEGRALS IN COMPLEX PLANE, THE SECOND ORDER DIFFERENTIAL EQUATIONS AT THE PARTIAL DERIVATIVES, THE FOURIER TRANSFORM AND SERIES; HE/SHE HAS THE CAPABILITY OF MAKING A CLEAR AND PUNCTUAL DEMONSTRATION OF THEOREMS. THE FINAL VOTE, EXPRESSED IN 30/30, IS OBTAINED FROM THE AVERAGE OF THE WRITTEN AND ORAL EXAMINATION VOTES.

Verification of learning

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

THE ORAL EXAMINATION IS AIMED AT DEEPENING THE LEVEL OF THE THEORETICAL KNOWLEDGE, AUTONOMY OF ANALYSIS AND JUDGMENT, AS WELL AS THE STUDENT'S EXHIBITION SKILLS.
THE LEVEL OF EVALUATION OF THE EXAMINATION TAKES INTO ACCOUNT THE EFFICIENCY OF THE METHODS USED, THE ACCURACY OF THE ANSWERS, AND CLARITY IN THE PRESENTATION.
THE MINIMUM LEVEL OF ASSESSMENT (18) IS ASSIGNED WHEN THE STUDENT DEMONSTRATES UNCERTAINTIES IN THE APPLICATION
OF THE METHODS FOR SOLVING THE PROPOSED EXERCISES AND HAS A LIMITED KNOWLEDGE OF THE MAIN THEOREMS UNDERLYING THE APPLICATIONS. THE MAXIMUM LEVEL (30) IS ATTRIBUTED WHEN THE STUDENT SHOWS A COMPLETE AND DEEP KNOWLEDGE OF THE RESOLUTION OF INTEGRALS IN COMPLEX PLANE, THE SECOND ORDER DIFFERENTIAL EQUATIONS AT THE PARTIAL DERIVATIVES, THE FOURIER TRANSFORM AND SERIES; HE/SHE HAS THE CAPABILITY OF MAKING A CLEAR AND PUNCTUAL DEMONSTRATION OF THEOREMS. THE FINAL VOTE, EXPRESSED IN 30/30, IS OBTAINED FROM THE AVERAGE OF THE WRITTEN AND ORAL EXAMINATION VOTES.

	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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