# Mathematical Methods of Physics

**Academic Year 2022/2023**- Teacher:

**Vito Claudio LATORA**

## Expected Learning Outcomes

Knowledge of the fundamentals of complex analysis, functional analysis and probability theory with applications to physics (specifically: quantum mechanics and statistical mechanics).

With reference to the topics covered in the teaching, the course will promote the following skills:

- Knowledge and understanding. Inductive and deductive reasoning skills. Ability to set up a simple problem using appropriate relationships between physical quantities (algebraic, integral or differential) and to solve it with analytical methods.

- Ability to apply knowledge and understanding. Ability to apply the knowledge acquired for the description of physical phenomena using rigorously the scientific method.

- Making judgments. Critical reasoning skills. Ability to identify the most appropriate methods to critically analyze problem data.

- Communication skills. Ability to present orally, with properties of language and terminological rigor, a scientific topic.

## Course Structure

Lectures (taught classes), and guided exercises (in class).

*Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes in line with the programme planned and outlined in the syllabus.*

## Required Prerequisites

## Attendance of Lessons

## Detailed Course Content

**PART I: Elements of complex analysis**

1) Complex numbers, operations and representations. De Moivre's formula and roots. Applications in physics.

2) Introduction to complex functions of complex variable. Domains, definitions and properties. Limits and continuity. Derivatives. Cauchy-Riemann and derivability conditions. Analytical functions. Singular points. Analytic functions and harmonic functions in physics. Exponential and logarithm functions. Trigonometric and hyperbolic functions.

3) Curvilinear integral. Cauchy's integral theorem. Primitive. Primitive theorem. Morera theorem. Integral formula of Cauchy. Integral Cauchy formula for derivatives.

4) Developments in series of complex functions. Convergence and criterion of the relationship. Weierstrass uniform convergence and criterion. Weierstrass theorem. Cauchy-Hadamard theorem. Taylor's theorem and Taylor series developments. Developments of elementary functions. Laurent theorem and Laurent series.

5) Residue method. Residues in poles of order m. Residue theorem. Improper integrals of rational functions. Integrals of rational trigonometric functions. Fourier integrals. Residue at infinity

**PART II: Elements of vector spaces and functional analysis **

1) Introduction to vector spaces. Linear differential equations. States of polarization of light. Vector spaces, definitions and properties. Bases and dimensions. Finite dimension vector spaces. Scalar product. Standard and distance based on a scalar product. Orthonormal bases.

2) Linear operators. Matrix representation of an operator. Space of polynomials. Hermite polynomials. Composition of two operators. Self-adjoint operators. Change of bases and unitary operators. Unitary transformations and similarity transformations.

3) Eigenvalues and eigenvectors. Autospace associated with an eigenvalue, and eigenvalue deneration. Secular equation. Own orthogonal matrices and rotations in R3. Diagonalization of Hermitian matrices. Fundamental theorem on the diagonalization of a self-adjoint operator. Applications: Linear dynamic systems. Coupled electrical circuits. Normal modes of vibration of the CO2 molecule. Projection operators. Operators functions.

4) Vector spaces of infinite dimension. D’Alembert equation. Normed spaces. Euclidean Spaces. Norm as a metric and completeness of a space. The L2 space. Hilbert spaces. Fourier theorem. Complete orthonormal systems and separable Hilbert spaces. Parseval identity. The space l2.

5) Operators in Hilbert spaces. Continuity and boundedness. Bounded operators and norm. Linear functionals. Riesz theorem. Adjoint, Hermitian and self-adjoint operators.

**PART III: Elements of probability theory**

1) Sample space and events. Definition of probability and its properties. Kolmogorov axioms. Inclusion-exclusion for two and more events. Sampling. Ordered and not ordered, with and without repetition. Binomial coefficients. Particles with spin 1/2. Birthday paradox.

2) Conditional probability. Conditional probability theorem. Independent and mutually independent events. Conditioned independence. Partitions and theorem of total probability. Conditioning technique. Bayes theorem. Confusion matrix

3) Random variables. Probability mass function, expectation value and test variance of Bernoulli, binomial, geometric and Poisson. Rare events and radioactive decays. Allergic reactions to a vaccine. Two or more random variables. Joint distribution and marginal distributions. Independent variables. Covariance and correlation coefficient. Conditional expectation values. Random walk.

4) Probability generating functions. Moments, centered moments and factorial moments. Generative functions of binomial, geometric and Poisson. Functions generating the sum of two variables.

5) Continuous random variables. Cumulative distribution and probability density. Uniform, exponential and normal distributions. Sum of n independent random variables. Central limit theorem.

## Textbook Information

C. Bernardini, O. Ragnisco, P.M. Santini, Metodi matematici della Fisica, Carocci Editore 1999

C. Presilla, Elementi di analisi complessa (Springer, Milano, 2014).

G. Cicogna, Metodi matematici della Fisica (Springer-Verlag, Italia 2008).

G. G. N. Angilella, Esercizi di Metodi Matematici della Fisica (Springer, Milano, 2011)

G. Fonte, Appunti di metodi matematici della fisica (Carocci, 2018)

CM Grinstead, JL Snell, Introduction to probability (Second revised Edition, American Mathematical Society 1997)

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Analytic functions | |

2 | Taylor and Laurent series, intgration by the method of residues | |

3 | Fourier series | |

4 | Hilbert spaces | |

5 | Linear operators and eigenvalue problems |

## Learning Assessment

### Learning Assessment Procedures

The exam will involve both solving exercises and theory questions on the topics of the course.

*Verification of the learning can also be carried out online, should the conditions require it.*

### Examples of frequently asked questions and / or exercises

Both the exercises and the theoretical questions will be on topics of complex analysis, on topics of vector spaces and functional analysis and on probability