MATHEMATICAL METHODS OF PHYSICS

International Teaching MATHEMATICAL METHODS OF PHYSICS

0522600017
DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
EQF7
PHYSICS
2021/2022

OBBLIGATORIO
YEAR OF COURSE 1
YEAR OF DIDACTIC SYSTEM 2021
PRIMO SEMESTRE
CFUHOURSACTIVITY
432LESSONS
224EXERCISES
Objectives
THE COURSE AIMS AT EXTENDING THE KNOWLEDGE OF THE MATHEMATICAL METHODS OF PHYSICS, ALREADY KNOWN FROM PREVIOUS STUDIES, TO MORE ADVANCED PROBLEMS SUCH AS THE THEORY OF DISTRIBUTIONS, PROBABILITY THEORY, THE PERTURBATIVE AND ASYMPTOTIC EXPANSIONS ETC... THIS KIND OF PROBLEMS ARE WIDESPREAD IN VARIOUS SECTORS OF THEORETICAL PHYSICS BUT THE ARGUMENTS DISCUSSED IN THE COURSE HAVE BEEN CHOSEN AS TO BE OF GENERAL INTEREST ALSO FOR THE EXPERIMENTAL AND PHENOMENOLOGICAL POINT OF VIEW.

KNOWLEGES AND COMPREHENSION:

THE COURSE PROVIDES ADVANCED MATHEMATICAL SKILLS REGARDING THE THEORY OF DISTRIBUTIONS, BASIC PROBABILITY THEORY, MORE ADVANCED PROBABILITY THEORY INCLUDING
LIMIT THEOREMS, LARGE DEVIATIONS AND STOCHASTIC PROCESSES. ALSO, PERTURBATIVE AND ASYMPTOTIC EXPANSIONS ARE CONSIDERED IN VARIOUS CONTEXTS (SOLUTIONS OF ORDINARY AND DIFFERENTIAL EQUATIONS, INTEGRAL CALCULUS ETC...)
Prerequisites
MATHEMATICAL COURSES FROM THE BACHELOR DEGREE. IN PARTICULAR: CALCULUS I, II, III, IV, GEOMETRY I AND II, MATHEMATICAL METHODS OF PHYSICS (BACHELOR LEVEL).
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.
Contents
DISTRIBUTIONS:
GENERAL DEFINITION, TEST FUNCTIONS AND THE SPACE OF TEST FUNCTIONS. DIRAC DELTA FUNCTION. (7 LESSON HOURS, 4 OF WHICH CLASSROOM-TAUGHT AND 3 OF EXERCISES AT THE BLACKBOARD CARRIED OUT BY THE LECTURER POSSIBLY WITH THE PARTICIPATION OF THE STUDENTS).

PROBABILITY THEORY: ASSIOMATIC AND FREQUENTIST DEFINITION
OF PROBABILITY, INDEPENDENCY, CONDITIONAL PROBABILITY, BAYES FORMULA, RANDOM VARIABLES, EXPECTATIONS, CONDITIONAL EXPECTATIONS, SUMS OF RANDOM VARIABLES, BINOMIAL DISTRIBUTION, POISSON DISTRIBUTION, PEARSON DISTRIBUTION, GAUSSIAN DISTRIBUTION, LOG-NORMAL DISTRIBUTION, GENERATING FUNCTION, DISTRIBUTION OF A FUNCTION OF A RANDOM VARIABLE, MARGINAL DISTRIBUTIONS, LIMIT THEOREMS, LAW OF LARGE NUMBERS, CENTRAL LIMIT THEOREM, LARGE DEVIATIONS, LEGENDRE TRANSFORM, CRAMER FUNCTION. STOCHASTIC PROCESSES. (23 LESSON HOURS, 13 OF WHICH CLASSROOM-TAUGHT AND 13 OF EXERCISES AT THE BLACKBOARD CARRIED OUT BY THE LECTURER POSSIBLY WITH THE PARTICIPATION OF THE STUDENTS).

ASYMPTOTIC EXPANSIONS: REGULAR AND SINGULAR PERTURBATIVE EXPANSIONS, ITERATIVE SOLUTIONS, NONDIMENSIONALIZATION, ORDER RELATIONS, THE O AND O SYMBOLS, ASYMPTOTIC AND CONVERGENT SERIES, STOKES PHENOMENON, ASYMPTOTIC EXPANSION OF INTEGRALS, RESONANT AND SECULAR TERMS, DUFFING EQUATION, METHOD OF STRAINED COORDINATES, MULTISCALE METHODS FOR SOLVING EQUATIONS. (26 LESSON HOURS, 15 OF WHICH CLASSROOM-TAUGHT AND 11 OF EXERCISES AT THE BLACKBOARD CARRIED OUT BY THE LECTURER POSSIBLY WITH THE PARTICIPATION OF THE STUDENTS).
Teaching Methods
THEORETICAL LESSONS WITH EXERCISES ON THE BLACKBOARD SOLVED BY THE TEACHER.
Verification of learning
ORAL EXAM AT THE END OF THE COURSE, WHOSE DURATION WILL BE ONE HOUR AT LEAST, WITH QUESTIONS CONCERNING ALL THE MAIN ARGUMENTS OF THE COURSE. IN PARTICULAR, THE STUDENT WILL BE CALLED TO DEMONSTRATE HOW DEEP IS HIS LEVEL OF UNDERSTANDING OF THE VARIOUS CONCEPTS AND PROBLEMS. THE STUDENT IS ASKED TO BE ABLE TO MAKE CONNECTIONS BETWEEN DIFFERENT PARTS OF THE PROGRAM AND TO SOLVE EXTEMPORANEOUS PROBLEMS. THESE ASPECTS ARE PARTICULARLY CONSIDERED FOR THE FINAL SCORE, BESIDES THE USUAL VERIFICATION OF THE KNOWLEDGE OF THE MATTER AND THE ABILITY TO PRESENT IT. THE MINIMUM MARK (18) IS EARNED BY A STUDENT WHO STUDIED MOST OF THE TOPICS OF THE COURSE BUT WHOSE COMPREHENSION IS NOT PARTICULARLY DEEPENED AND WHOSE CAPACITY TO CONNECT DIFFERENT PARTS IS NOT WELL DEVELOPED. THE MAXIMUM MARK (39) IS GIVEN TO STUDENTS WITH A TOTAL AND DEEP CONTROL OF THE ARGUMENTS OF THE COURSE. THE LAUDE IS GIVEN WHEN THESE CAPACITY IS OWNED AT AN OUTSTANDING LEVEL. .
Texts
NOTES OF VARIOUS AUTHORS AS ITEMIZED BELOW, THEY ARE DISTRIBUTED IN THE CLASS OR SHARED ON THE INTERNET WITH THE STUDENTS AT THE BEGINNING OF THE COURSE.

S. DE SIENA, AN INTRODUCTION TO HILBERT SPACES (WITH EXERCISES AND COMPLEMENTS) - NOTE

A.CRISANTI, SVILUPPI ASINTOTICI, NOTE.

M.FALCIONI E A.VULPIANI: NOTE INTRODUTTIVE SULLA TEORIA DELLA PROBABILITÀ


OTHER TEXTBOOKS FOR CONSULTATION AND DEEPENING SOME SUBJECTS:

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.
More Information
STUDENTS ARE ENCOURAGED TO KEEP IN CONTACT WITH THE TEACHER AT ANY TIME FOR FURTHER CLARIFICATIONS.
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