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Lecturer
Serena Cenatiempo, serena.cenatiempo@gssi.it
Timetable and workload
Lectures: 20 hours
Course description
This course aims at providing a mathematical introduction to the theory of Quantum Mechanics and to many particle problems in quantum mechanics, as a preparation to follow research seminars and advanced courses on this topic. After to a short introduction about the formulation of Quantum Mechanics and the characteristic features of the new theory compared to classical physics, we will discuss the study of systems made up by a large number of quantum particles. While in the absence of interaction, the properties of many body systems can be deduced from the single particle Hamilton operator, in presence of interactions one needs to study the full N particle Schrödinger equation, with N very large and virtually infinite. This requires the development of new mathematical methods and tools.
Course requirements
Basic elements of the theory of linear operators in Hilbert spaces.
Lebesgue decomposition theorem for measures.
Course content
1 (Nov. 27): A historical introduction
2 (Nov. 28): The postulates of Quantum Theory
3 (Dec. 4): One particle models: the free particle
4 (Dec. 5): One particle models: the harmonic oscillator
5 (Dec. 7): The spectrum of the hydrogen atom. Pure and mixed states.
6 (Dec. 11): Entanglement. Many-body Hamiltonians. Bosons&fermions. Reduced densities
7 (Dec. 12): Statistical Ensembles in Quantum Mechanics
8 (Dec. 18): Creation and annihilation operators. The free Bose & Fermi gases
9 (Dec. 19): Bose-Einstein condensation for interacting bosons: statics and dynamics
10 (Dec. 21): A proof of Bose-Einstein condensation in the mean field regime
References
One-particle Quantum Mechanics
Quantum Statistical Mechanics
Entanglement