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Lecturers
Giulia Basti, giulia.basti@gssi.it
Serena Cenatiempo, serena.cenatiempo@gssi.it
Timetable and workload
Lectures: 18 hours
In addition to those hours TA sessions may be organized to recall some of the mathematical background.
Course description and outcomes
The class aims to discuss the mathematical problems related to the study of the collective properties of many particle systems, in regimes where the classical particle behaviour is no longer appropriate, and one should rather consider the formalism of Quantum Mechanics. No a priori knowledge in quantum mechanics will be assumed. On the contrary we will give an introduction to the theory starting from its historical and experimental motivations, and then apply it to some simple and paradigmatic one-particle models, before addressing the more challenging goal of investigating the collective behaviour of large quantum systems.
Course requirements
Basic elements of the theory of linear operators in Hilbert spaces.
Course content
Lec.1 - Historical introduction [slides]
Lec.2 - Building the formalism of the theory [screen notes]
Lec.3 - Spectral theorem for self-adjoint operators & unitary groups [screen notes]
Lec.4 - Unitary evolution operator & Free particle [screen notes]
Lec.5 - Harmonic oscillator & classical limit of Quantum Mechanics [screen notes]
Lec.6 - Hydrogen atom [screen notes]
Lec. 7 - Quantum Many-Particle Systems [screen notes]
Lec. 8 - Statistical Ensembles [screen notes]
Lec. 9 - Non-Interacting Particles [screen notes]
TA classes:
TA 1 - Self-adjoint operators [screen notes]
TA 2 - Spectrum, resolvent identities, trace-class operators [screen notes]
References
E.H.Lieb and R.Seiringer. The Stability of Matter in Quantum Mechanics. Cambridge, 2009.
R. Seiringer. Cold Quantum Gases and Bose-Einstein Condensation. Lecture notes from the school ``Quantum Theory from Small to Large Scales'', August 2-27, 2010 [available here]
F. Schwabl. Statistical Mechanics. Springer, 2006.
A. Teta. A Mathematical Primer on Quantum Mechanics. Springer, 2018.
Examination and grading
This class comes without any final exam, however written exercises will be suggested to the students during the course.