The course presents the key mathematical tools needed for the study of quantum systems.
This course introduces the fundamental concepts of quantum mechanics, providing students with the mathematical techniques necessary for a rigorous formulation of the theory. In particular, it focuses on the algebraic structure of quantum observables and analyzes the theorems essential to the representation of this algebra. Finally, some tools from operator theory and Hilbert space analysis will be used to derive the evolution equations of Schrödinger and Heisenberg and to discuss their solutions.
Starting from few basic concepts from quantum mechanics, which will be recalled in the first lectures, the student will learn the tools that play a crucial role in the formalization of quantum mechanics as a rigorous mathematical theory. In order to pursue this goal, the student will
With these results the student acquires the ability to switch from an abstract algebraic approach to a more concrete one, based on the theory of operators on a Hilbert space, thus clarifying the relation with the traditional description of quantum mechanics. More specifically, the student will learn how to
Towards the end of the course, in order to deal with applications to concrete problems of physical interest, the student will explore some tools from the theory of unbounded self-adjoint operators on Hilbert spaces.
There are no specific prerequisites.
Taught class.
Preliminary physical observations
Algebraic description of a physical system
Quantum systems and non-commutativity
Quantum particle
Schrödinger equation
Examples and applications
Lecture notes, as well as additional references, will be made available during the course.
Ricevimento: By appointment.
From February 23rd, 2026.
The exam consists of an oral test. The test is passed with a minimal result of 18/30. During the exam session students can schedule an appointment for an oral test by getting in touch directly with the lecturer.
The exam consists of an oral test, during which the student is asked to demonstrate familiarity with the concepts and the tools presented during the lectures. In particular, the student will present some of the theorems discussed during the lectures, along with the relevant definitions, and reproduce the respective proofs autonomously.
Contact the lecturer for other information not included in the teaching schedule.