Normed spaces and operators on them. Sketches of Lebesgue integration. Hilbert spaces and operators. Spectral theory for self-adjoint operators on Hilbert spaces. Applications to Quantum Mechanics. Representations of the canonical commutation relations and Weyl algebra. Stone’s theorem and Hamiltonian operator. Harmonic oscillator. Sketches of the representation theory of matrix Lie groups and algebras. Angular momentum and spin. Kato-Rellich’s theorem. Self-adjointness and spectrum of the hydrogen’s atom Hamiltonian.
The aim of the course is to deepen the mathematical analysis knowledeges necessary for the conceptually clear formulation of physical theories and related mathematical problems, with particular attention to the formulation of the mathematical foundations of quantum mechanics.
KNOWLEDGE AND UNDERSTANDING:
At the end of the course, the student will be able to understand and describe the fundamental results of functional analysis, and in particular of the theory of normed spaces, of Lebesgue integration, of Hilbert spaces and self-adjoint operators on them.
APPLYING KNOWLEDGE AND UNDERSTANDING:
At the end of the course, the student will be able to apply the basic results of functional analysis to the mathematically rigorous formulation and resolution of fundamental mathematical problems of quantum mechanics such as the analysis of the representations of the canonical commutation relations, the harmonic oscillator, angular momentum, spin, and the hydrogen atom.
The student must be able to critically discuss the links between the concepts learned, identifying the fundamental logical connections and possible variants, as well as analyzing a mathematical problem inherent to the course topics, and choosing the most suitable and convenient methodology, in a motivated manner, for its solution.
The student must be able to communicate in a clear and coherent way, both synthetically and analytically, the definitions, the theorems and related proofs, highlighting the relevant hypotheses and the crucial steps, properly using the formal language of functional analysis.
At the end of the course, the student will be able to read and understand advanced textbooks and partially research papers of a mathematical physics subject inherent to the topics of the course in order to be able to study them independently.