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 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.

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 group.
• 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.