MATHEMATICAL METHODS FOR PHYSICS

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

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

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.

Elements of 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. Herpes 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 with infinite dimension. Equation of D’Alembert. 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.

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.

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)