OVERVIEW

The course presents the key mathematical tools needed for the study of quantum systems.

AIMS AND CONTENT

LEARNING OUTCOMES

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.

AIMS AND LEARNING OUTCOMES

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

• analyze the natural algebraic structure carried by the set of quantum observables and
• investigate the foundational theorems about representations of the algebra of quantum observables, which play a crucial role in the analysis of quantum systems.

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

• model a quantum particle through its Weyl algebra and the associated Schrödinger representation and
• describe the dynamics of a quantum system through the Schrödinger equation.

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.

PREREQUISITES

There are no specific prerequisites.

TEACHING METHODS

Taught class.

SYLLABUS/CONTENT

Preliminary physical observations

• Crysis of classical physics at the atomic scale.

Algebraic description of a physical system

• Classical hamiltonian systems; states and observables.
• Observables as a C*-algebra.
• Mathematical theory of C*-algebras (both in the commutative case and in general).

Quantum systems and non-commutativity

• Heisenberg principle and non-commutativity.
• Quantum states and the Gelfand-Neimark-Segal (GNS) representation theorem.

Quantum particle

• Weyl algebra and Heisenberg commutation relations.
• Von Neumann uniqueness theorem.
• Construction of the Schrödinger representation.
• Gaussian states.

Schrödinger equation

• Time-evolution automorphisms and their representation (Heisenberg).
• Free quantum particle.
• Unbounded self-adjoint operators.

Examples and applications

• Superposition principle.
• Quantum harmonic oscillator.
• Quantum particle in a potential well.
• Hydrogen atom.

RECOMMENDED READING/BIBLIOGRAPHY

Lecture notes, as well as additional references, will be made available during the course.

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

From February 23rd, 2026.

EXAMS

EXAM DESCRIPTION

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.

ASSESSMENT METHODS

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.

Exam schedule

FURTHER INFORMATION

Contact the lecturer for other information not included in the teaching schedule.