Advanced Mechanics

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/07 Mathematical Physics.

Examination method

Oral exam.

Learning
objectives

The course aims to provide students with the fundamental concepts, models and mathematical tools of Quantum Mechanics, starting from the Hamiltonian formulation of Classical Mechanics. In this way, the course seeks to provide basic methodologies and skills while at the same time stimulating curiosity and motivation for further study in this area of Mathematical Physics.

Syllabus

Review of Hamiltonian mechanics: Poisson brackets; canonical transformations; Hamilton-Jacobi equation; action-angle variables, adiabatic invariants for a particle in a Coulomb potential. Bohr-Sommerfeld-Ehrenfest quantisation rules. Matrix formulation of quantum mechanics (Heisenberg-Born-Jordan). Heisenberg commutation rules and operator formulation (Dirac). Stern-Gerlach experiment. Pure states, mixed states. Representations, unitary transformations, dynamics in the Heisenberg and Schroedinger pictures and their equivalence. Schrödinger equation. Stationary states. Position and momentum representations. Heisenberg uncertainty relations. Diffraction and interference. 1-dim problems. Closed, Hermitian and (essentially) self-adjoint operators and related criteria (spectrum and spectral function). Spectral decomposition and eigenrepresentations of x,p. One-parameter groups of unitary operators, theorems of Stone and von Neumann. Kato and Kato-Rellich theorems, and stability of atomic systems. Complete sets of commuting observables. Representations of the angular momentum algebra; spherical harmonics. The hydrogen atom.

Expected learning
outcomes

On completion of the course, students must demonstrate that they

understand and know the general issues relating to the mathematical modelling of systems in the quantum setting;
can apply the theoretical knowledge acquired to understand and solve some simple models (essential algebraic-analytical methodologies and skills) in the field of Quantum Mechanics;
can communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
can identify the most appropriate methods to analyse and solve a problem relating to the course topics and interpret the results correctly.

Learning outcomes
to be assessed

Correctness, completeness and clarity in the oral presentation of the course topics. Ability to carry out the mathematical steps needed to prove theorems or to obtain quantitative results in models.

Advanced Mechanics