1) Knowledge and understanding: comprehension of the physical foundations, of the statistical interpretation and of the ensuing formal structure of quantum mechanics. 2) Applying knowledge and understanding: application of the quantization techniques of basic physical systems like n-level systems, harmonic oscillators, hydrogen atom and of their perturbations. 3) Making judgements: comprehension and application of the consequences of quantization in microphysical contexts. 4) Communication skills: ability to motivate and explain the consequences of the statistical structure of quantum mechanics. 5) Learning skills: ability of using the knowledge of the formal apparatus and of the basic tools of quantum mechanics to study and understand elementary particle physics, quantum field theory, solid state physics, the statistical physics of many-body systems and rapidly developing field of quantum technologies.

Hamiltonian Mechanics and Linear Algebra

Aim of the course is to provide the students with the physical motivations at the roots of Quantum Mechanics, with its formal apparatus and with the main applications to microphysical contexts. The course consists of four parts: the first one will justify and develop the mathematical techniques of Quantum Mechanics comparing them with those of Classical Physics. In the second part there will be presented and discussed the postulates of Quantum Mechanics e the laws of quantum dynamics. The third part will present the quantization of position and momentum, of the free-particle dynamics, of the harmonic oscillator and of systems of coupled harmonic oscillators, of the angular momentum and of the hydrogen atom. In the last part perturbative techniques will be introduced as those pertaining to non-degenerate and degenerate energy spectra and to time-dependent potentials.

M. Le Bellac, Quantum Physics, Cambridge University Press 2006

1. Hilbert spaces and linear operators 1.1 Projections, density matrices, probabilities and mean values 1.2 Self-adjoint operators and spectral theorem 1.3 Linear algebra and applications to finite-level quantum systems 2. Postulates of Quantum Mechanics 2.1 States and observables 2.2 Uncertainty relations 2.3 Dynamics: Schrödinger and Heisenberg pictures 3. Quantization 3.1 Angular momentum and spin 3.2 Harmonic oscillator: creation and annihilation operators 3.3 Hydrogen atom 4. Perturbation Theory 4.1 Time-independent perturbation: non-degenerate case 4.2 Time-independent perturbation: degenerate case 4.2 Time-dependent: Fermi golden rule.

Blackboard lectures with discussions and solutions of illustrative problems aiming at possibly interactively involving the students.

Only written exam consisting of three problems, each one evaluated with a total maximum of 10 marks and each one subdivided into three logically correlated exercises of varying difficulty with evaluation up to a maximum of five marks. The exercises aim at ascertaining whether the comprehension of the formal structure of quantum mechanics has been reached together with its consequences for the description of microscopic phenomena. The three exercises into which the three problems are subdivided check that the minimal knowledge has been reached and evaluate the ability to apply them to physical context of increasing difficulty. The maximum mark, 30 cum laude, is attributed when all three exercises of the three problems have been correctly solved in the most direct and clear possible way or revealing particular insights on the part of the students.