Probability and Statistical Mechanics. Part 1: Quantum Mechanics

by Prof. Serena Cenatiempo (GSSI), Dr Daniele Ferretti (GSSI)

Tuesday 5 Nov 2024, 09:00 → Friday 29 Nov 2024, 11:00 Europe/Rome

Part 1 of the pillar course in Statistical Mechanics course will introduce the mathematical foundations of Quantum Mechanics. Beginning with key paradigmatic one-particle problems, we will then focus on the mathematical analysis of large systems of interacting quantum particles.

Timetable     
The course consists of regular lectures by Prof. Serena Cenatiempo and TA sessions led by Dr. Daniele Ferretti, with three lectures scheduled each week as follows:

• Mon, Thu 8.30-10.30 
• Tue 10.30-12.30      
   

Course description      
Quantum statistical mechanics deals with the study of quantum systems made up by a large number of particles. In particular it aims to understand the macroscopic behaviour of these systems, starting from their microscopic fundamental description. 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. 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.      
 

Course requirements      
Basic elements of the theory of linear operators in Hilbert spaces.
 

Course content      
A tentative schedule of the course is detailed below.      

Nov. 4:     The postulates of Quantum Theory       
Nov. 5:     TA: Self-adjoint operators       
Nov. 7:     One particle models: Harmonic Oscillator     

Nov. 11:   TA: Spectrum and Spectral Theorem for self-adjoint operators     
Nov. 12:   One particle models: Free particle     
Nov. 14:   TA: Spin     

Nov. 18:   One particle models: Hydrogen Atom     
Nov. 19:   Pure and Mixed States. Entanglement.      
Nov. 21:   Many particle systems in quantum mechanics     

Nov. 25:   Statistical Ensembles in Quantum Mechanics     
Nov. 26:   TA: Second Quantization     
Nov. 27:   Non interacting Bose and Fermi gases

Three optional classes on Bose-Einstein condensation in interacting Bose gases and the rigorous derivation of the Gross-Pitaevskii equation will be offered in February 2025.


Lecture Notes

• D. Ferretti. Elements of Mathematical Foundations of Quantum Mechanics. 
• G. Basti and S. Cenatiempo. An invitation to Quantum Statistical Mechanics. 
• 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.

References

• E.H.Lieb and R.Seiringer. The Stability of Matter in Quantum Mechanics. Cambridge, 2009.
• F. Schwabl. Statistical Mechanics. Springer, 2006.
• A. Teta. A Mathematical Primer on Quantum Mechanics. Springer, 2018.

Elements_of_Mathematical_Foundations_of_Quantum_Mechanics.pdf