Mathematical Methods of Physics

Knowledge of the fundamentals of complex analysis and functional analysis with applications to physics.

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.

The course is based on 6 CFU (50 hours) of taught classes.

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.
 

PART I: Elements of complex analysis (25 hours)

1) Complex numbers, operations and representations. De Moivre's formula and roots. Applications in physics. (2 hours)

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. (5 hours)

3) Curvilinear integral. Cauchy's integral theorem. Primitive. Primitive theorem. Morera theorem. Integral formula of Cauchy. Integral Cauchy formula for derivatives. (6 hours)

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. (6 hours)

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. (6 hours)

PART II: Elements of vector spaces and functional analysis (25 hours)

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 hours)

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. (5 hours)

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. (6 hours)

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. (6 hours)

5) Operators in Hilbert spaces. Continuity and boundedness. Bounded operators and norm. Linear functionals. Riesz theorem. Adjoint, Hermitian and self-adjoint operators. (6 hours)

1. C. Bernardini, O. Ragnisco, P.M. Santini, Metodi matematici della Fisica, Carocci Editore 1999
2. C. Presilla, Elementi di analisi complessa (Springer, Milano, 2014).
3. G. Cicogna, Metodi matematici della Fisica (Springer-Verlag, Italia 2008)
4. G. Fonte, Appunti di metodi matematici della fisica (Carocci, 2018)
5. G. G. N. Angilella, Esercizi di Metodi Matematici della Fisica (Springer, Milano, 2011)

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.

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

1. Proof of theorems on analytic functions 
2. Characterization of singularities. Laurent series 
3. Cauchy integral formula for derivatives 
4. Integration of complex functions
5. Residue theorem and calculation of various types of integrals 
6. Normed spaces, Euclidean spaces, Hilbert spaces 
7. Changes of basis and unitary operator.
8. Continuous operators. Norm of an operator
9. Function of an operator. Projection operators. 
10. L2 space: definitions and properties. l2 space, definitions and properties. 
11. Continuous linear functionals. Riesz theorem.