# QUANTUM MECHANICS

## Learning outcomes of the course unit

By means of frontal lessons, the student acquires the

methods and knowledges required to describe elementary

physical systems using the theory of Quantum Mechanics.

Through practical classroom exercises connected to

some important topics, students learn how to

apply the acquired knowledge using mathematical calculus.

## Prerequisites

Physics II, Mathematical Methods of Physics

## Course contents summary

The course aims to provide the students with the general elements of Quantum Mechanics. Therefore,the first part of the course deals with the physical effects and the experiments that led to the formulation of the theory. In the second part we elaborate the theory, introducing the mathematical formalism solving some relevant physical system. in the third part the theory is applied to some problem of atomic physics.

## Course contents

1) Introduzione

Black-body and UV catastrophe, Planck idea, specific heat, photoelectric effect, Bohr atom, Bohr-Sommerfeld. Compton effect and DeBroglie.

2) Quantum Mechanics

Schrodinger equation and properties, free solution, one-dimensional case. Probabilistic interpretation of wave function.

General formalism: Hilbert spaces , states, observables/operators,

probabilities, expectation values medi, complete set of observables, measures,Heisenberg relations. One-dimensional systems: well, tunnelling, transmission and reflections, scattering. Harmonic oscillator

(analytic and algebric solution). Angular orbital momentum, spherical harmonics a. Rotations and symmetries, unitary transformations. Three-dimensional systems: rotational invariance, central potentials, hydrogen atom. Introduction of magnetic field, gauge symmetry, Landau levels. General theory of angular momentum: algebraic solution, spin and sum of angular momentums. Time-independent perturbation theory: non-degenerate and degenerate cases.

Time-dependent perturbation theory: general formalism, transition amplitudes, Fermi golden rules.

3) Application to atomic physics.

Variazional method and applications. Spin-statistics, Pauli principle, bosons e fermions, applications to multi-electron atoms, helium atom. Atoms in magnetic field: Zeeman effetcs, transitions. Fine structure: relativistic corrections, spin-orbit etc.. Comparison with exact Dirac solution. Hyperfine structure. Self-consistent methods: Hartree-Fock and generalizations (general formalism).

## Recommended readings

Istituzioni di fisica teorica, Enrico Onofri, Claudio Destri, Editor: Carocci

Meccanica quantistica moderna; Jun J. Sakurai,Jim Napolitano, Editor: Zanichelli

## Teaching methods

The didactic activities are composed

of frontal lessons alternating with exercises.

## Assessment methods and criteria

Verification of the knowledge takes place

through a written test based on three exercises, lasting 3 hours. The oral examination takes 3 questions on the 3 part of program.