# MATHEMATICAL METHODS IN QUANTUM MECHANICS

## AIMS AND CONTENT

LEARNING OUTCOMES

In this course will be presented the basic concepts of quantum mechanics, highlighting the mathematical techniques necessary for the strict formalization of this theory. In particular, the algebraic structure of quantum observables will be studied and the theorems necessary for the representation of this algebra will be analyzed. Finally, some instruments of operator theory and analysis of Hilbert spaces 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 course will emphasize the mathematical tools that play a crucial role in the formalization of quantum mechanics as a rigorous mathematical theory. In order to pursue this goal, it will be necessary to examine the canonical algebraic structure carried by the set of quantum observables. This point of view naturally guides us to foundational theorems in the representation theory of the algebra of quantum observables. Furthermore, it helps us unravelling the main features of such representations. Eventually, these results lead to a change of perspective from an abstract algebraic approach to a more concrete one, based on the theory of operators on a Hilbert space, where a concrete description of the quantum particle and of its dynamics through the Schrödinger equation becomes explicitly tractable. Towards the end of the course, in order to deal with applications to concrete problems of physical interest, it becomes necessary to develop some tools from the theory of selfadjoint unbounded operators on Hilbert spaces.

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.

**The quantum particle**

- Weyl algebras and Heisenberg group.
- Von Neumann uniqueness theorem.
- Construction of the Schrödinger representation.
- Gaussian states.

**Schrödinger equation**

- Time-evolution automorphisms and their representation (Heisenberg).
- The 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

References will be provided during the course.

## TEACHERS AND EXAM BOARD

**Ricevimento:** By appointment.

Exam Board

PIERRE OLIVIER MARTINETTI (President)

NICOLA PINAMONTI

MARCO BENINI (President Substitute)

CLAUDIO BARTOCCI (President Substitute)

## LESSONS

Teaching methods

Taught class.

## EXAMS

Exam description

The exam consists of an oral test, during which the student is asked to demonstrate familiarity with concepts and tools presented in the lectures. More specifically, the studente is expected to have learnt definitions and theorems exhamined during the lectures and to be able to prove them autonomously.

Assessment methods

The exam consists of an oral test. Please get in touch with the teacher to schedule an appointment.

## FURTHER INFORMATION

Attendance is recommended for this course.