FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

Fisica FUNDAMENTALS OF MATHEMATICAL METHODS FOR PHYSICS

0512600031
DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
EQF6
PHYSICS
2022/2023

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2017
SPRING SEMESTER
CFUHOURSACTIVITY
864LESSONS
112EXERCISES
ExamDate
APPELLO DI FEBBRAIO 202317/02/2023 - 10:00
Objectives
THE COURSE PROVIDES AN ADVANCED MATHEMATICAL KNOWLEDGE NECESSARY FOR THE COMPREHENSION AND DESCRIPTION OF THE MORE RELEVANT PHYSICAL PHENOMENA OF MODERN PHYSICS.
KNOWLEDGE AND COMPREHENSION ABILITY:
THE COURSE GIVES TO THE STUDENTS MATHEMATICAL KNOWLEDGE AND TOOLS BASED ON THE CONCEPT OF HILBERT SPACE AND OPERATORS, AT THE BASIS OF QUANTUM MECHANICS. MOREOVER PROVIDES COMPETENCES ON COMPLEX ANALYSIS AND FOURIER ANALYSIS AND GIVES THE PROPER INSTRUMENTS TO SOLVE EXCERCISES AND PROBLEMS IN THE FRAMEWORK ABOVE.

APPLYING KNOWLEDGE AND COMPREHENSION:
THE STUDENT WILL ACQUIRE A DEEP COMPREHENSION OF THE STUDIED TOPICS THANKS TO WHICH HE WILL BE ABLE TO RECOGNIZE THE MATHEMATICAL STRUCTURE BELOW THE DESCRIPTION OF THE QUANTUM PHENOMENA AND ON THE OTHER SIDE WILL BE ABLE TO RECOGNIZE THE RELEVANCE OF THE APPLICATIONS IN PHYSICS OF THE MAIN CONCEPTS OF COMPLEX ANALYSIS BASED ON THE USE OF FOURIER AND LAPLACE TRANSFORM, CONFORMAL TRANSFORMS.


JUDGMENT AUTONOMY:
KNOWING HOW TO IDENTIFY THE MOST APPROPRIATE METHODS TO SOLVE ADVANCED PHYSICS PROBLEMS IN FIELDS
SUCH AS FLUID DYNAMICS AND ELECTRODYNAMICS.

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 GENERAL PHYSICS COURSES.
Contents
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. [18H; EXC. 10H]

LINEAR VECTORIAL SPACES; INTRODUCTION TO MEASURE THEORY AND LEBESGUE INTEGRALS; L² FUNCTIONS SPACE. [12 H; EXC. 4H]

FOURIER FOURIER SERIES AND TRANSFORMS IN L1 AND L2. 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. [16H; EXC. 8H]

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

HILBERT SPACES: DEFINITION AND PROPERTIES, ORTHONORMAL BASIS. DEFINITION OF OPERATORS (HERMITEAN, UNITARY, PROJECTION). EIGENVALUES AND EIGENVECTORS OF AN OPERATOR. CHANGE OF BASIS. SPECTRAL REPRESENTATION OF AN OPERATOR. HILBERT SPACE C^2: THE BLOCH SPHERE, PAULI MATRICES, TWO-LEVEL SYSTEMS, OBSERVABLES AND INTRODUCTION TO QUANTUM COMPUTING [16H].


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.
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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