ADVANCED QUANTUM MECHANICS

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.

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.

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.