96754 - Mathematical Methods of Quantum Mechanics

Learning outcomes

At the end of the course the student gets acquainted with some of the modern mathematical techniques used in quantum mechanics, such as abstract spectral theory, semiclassical pseudodifferential calculus, microlocalization. He is able to apply them in order to study typical phenomena in quantum mechanics, such as the tunnel effect and quantum resonances.

Course contents

-) General introduction to quantum phenomenology and the elementary formalism in finite dimension.

-) A first infinite-dimensional extension and notion of "quantization".

-) Uncertainty principle. 

-) Definition and characterization of quantum observables and quantum states in infinite-dimensional Hilbert space by means of abstract spectral theory.

-) Non-boolean logic of quantum mechanics, states as generalized probability measures and Gleason’s theorem.

-) Quantum symmetries in terms of unitary projective representations of groups, Wigner and Kadison theorems.

-) Overview about the abstract algebraic formulation of quantum theories.

Readings/Bibliography

Main references:

-) J. von Neumann. Mathematical foundations of Quantum Mechanics (Princeton University Press)

-) V. Moretti. Spectral Theory and Quantum Mechanics (Springer)

-) F. Strocchi. An Introduction to the Mathematical Structure of Quantum Mechanics (World Scientific)

-) Brian C. Hall. Quantum Theory for Mathematicians (Springer)

-) Lecture notes

Teaching methods

Classroom lectures

Assessment methods

Oral examination

Office hours

See the website of Davide Pastorello

See the website of Giacomo De Palma