MATHEMATICAL METHODS OF PHYSICS WITH EXERCISES MOD. II
- Overview
- Assessment methods
- Learning objectives
- Contents
- Full programme
- Bibliography
- Teaching methods
Basic elements of linear algebra, and of the theory of finite dimensional vector spaces
Written exam in which students must solve three exercises, each divided by two to four issues, of the same type as those performed in the classroom.
An oral test follows in which the student must demonstrate to have reached a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use, beyond the mere formal manipulation.
It is expected that the students of this course will acquire a certain operational familiarity with those mathematical tools which, although they have long been commonly used in quantum mechanics, are nevertheless advanced enough to exceed the limits of basic courses in mathematical analysis; nevertheless maintaining a degree of awareness of the logical articulation of the underlying mathematical theory, sufficient to allow a critical use of it, beyond the pure and simple formal manipulation.
The course consists of a synthetic but logically coherent introduction to the elements of Functional Analysis on which the mathematical formalism of quantum mechanics is based. The traditional elementary theory of Hilbert spaces will be accompanied by an "effective" introduction to the theory of temperate distributions. Without renouncing logical coherence, the use of abstract aspects of the theory of topological vector spaces will be minimized, while the emphasis will be placed on the most frequently used elementary distributional techniques, such as the Fourier transform, and Green functions.
General notion of functional space. Topological vector spaces. Normed and pre-Hilbertian spaces, Cauchy-Schwarz inequality, parallelogram identity, polarization identity. Banach and Hilbert spaces. Spaces of sequences. Synthetic introduction to the abstract measure theory and integration theory. Taking limit under the integral sign. Quadratic mean convergence. Spaces of square summable functions. Subspaces of a Hilbert Space; Theorem of projections. Orthogonal decomposition. Orthonormal systems and Hilbertian bases. Generalized Fourier series. Separable spaces. Hilbertian isomorphism; Continuous linear operators. Representation of continuous linear operators. Algebra of continuous operators: unitary operators, projectors, adjoint operators; convergence of series of operators. Fourier series for periodic functions. Trigonometric series. Fourier integral; elementary properties, the Riemann-Lebesgue lemma. Fourier transform in the Schwarz space of fast decreasing test functions. The Hermite basis and its basic properties. The Fourier-Plancherel transform. Tempered distributions as weak limit of square summable functions. Regular and singular distributions. The Dirac’s delta. Distributional derivatives. The Principal Part P1/x distribution. The pointlike charge potential. Further operations: change of variables, product, tensor product. The Fourier transform of tempered distributions; computational rules; explicit examples. Convolution of distributions. Fundamental solutions of a linear differential operator. Fundamental solutions, and the Cauchy problem; Green functions. Fundamental solutions for the diffusion equation, the Schroedinger equation for the free particle, and the wave equation.
Written notes provided by the teacher during the course
Frontal lessons and exercises