# ADVANCED QUANTUM MECHANICS

**Academic Year 2017/2018**- 1° Year - Curriculum ASTROPHYSICS, Curriculum PHYSICS APPLIED TO CULTURAL HERITAGE, ENVIRONMENT AND MEDICINE, Curriculum CONDENSED MATTER PHYSICS, Curriculum NUCLEAR AND PARTICLE PHYSICS, Curriculum THEORETICAL PHYSICS and Curriculum NUCLEAR PHENOMENA AND THEIR APPLICATIONS

**Teaching Staff:**

**Vincenzo GRECO**

**Credit Value:**6

**Scientific field:**FIS/02 - Theoretical physics, mathematical models and methods

**Taught classes:**35 hours

**Laboratories:**15 hours

**Term / Semester:**1°

## Learning Objectives

Supply an advanced knowledge on quantum theory including its relativistic extension and its applications to the study of physical systems in modern physics. Introduce the

student to the formulation of Quantum Mechanics in terms of Path Integrals and to the quantization of the electromagnetic field.

## Detailed Course Content

**Approximation Methods**

Overview of Time-Independent Perturbation Theory; Interaction (or Dirac’s) representation of quantum mechanics; time dependent perturbation theory (instantaneous, periodic, adiabatic); Fermi Golden Rule; Applications to the interaction with classical electromagnetic field and photoelectric effect; Berry's geometrical phase; WKB method and applications to Bohr-Sommerfeld quantization, finite double well potential and tunneling processes; Eikonal approximation; exercises.

**Scattering Theory**

Lippmann-Schwinger equation; Scattering amplitude and differential cross section; Born approximation; Expansion in partial waves and phase shifts; Low energy scattering and bound states; Elastic and inelastic scattering; Inelastic electron-atom scattering and form factors; Resonant scattering and Feynmann diagrams for non-relativistic interacting systems; exercises.

**Second Quantization**

Identical particles, many-particle states and the formulation of quantum mechanics in second quantization; Symmetric and anti-symmetric states: bosons and fermions; One-body and two-body operators; Two-particle correlation functions for bosons and fermions; Hanburry-Twiss effect; weakly interacting Bose gas and Gross-Pitaevskii equation.

**Primer of Quantum Theory for the electromagnetic field**

Schroedinger equation in a external e.m. field and gauge invariance; Landau levels; Bohm-Ahranov effect and magnetic monopole; simplified approach to the quantization of electromagnetic field; spontaneous radiative emission and dipole transitions; exercises.

**Path-Integrals**

Propagators and Green-functions; Path-Integral formulation of quantum mechanics; Examples: free particle, harmonic oscillators; primer on instantons.

**Relativistic Quantum Mechanics**

Klein-Gordon Equation and Klein’s paradox; Casimir effect; Dirac Equation and the free particle and anti-paticle solutions; Weyl and Majorana representations; Non-relativistic reduction of Dirac equation: Pauli equation; Charge, Parity and Time reversal simmetries; Dirac particle in a Coulomb field; hyperfine structure and Lamb-shift; exercises.

## Textbook Information

1) Giuseppe Nardulli - Meccanica quantistica: applicazioni, vol II, Ed. Franco Angeli.

2) J.J. Sakurai, Advanced Quantum Mechanics, Ed. Addison-Wesley.

3) J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Ed. Addison-Wesley.

4) J.D. Bjorken and S. D. Drell - Relativistic Quantum Mechanics, Ed. McGraw-Hill.

5) P. Roman, Advanced Quantum Theory, Ed. Addison-Wesley.