# ADVANCED QUANTUM MECHANICS

**Academic Year 2022/2023**- Teacher:

**VINCENZO GRECO**

## Expected Learning Outcomes

The teaching proposes to provide a knowledge of quantum mechanics including its relativistic extension. In particular, the objective is to provide a knowledge of the main methods for understanding the quantum behavior of the physical systems of interest for modern physics, explicitly deriving the time-dependent pertubatical theory and the general elements of the quantum approach to the scattering process. Moreover, the teaching will allow to access to the more advanced formulations of quantum mechanics such as quantization of the electromagnetic field, the formulation of quantum mechanics in terms of Feynmann integrals and the relativistic formulation of quantum mechanics with the Dirac and Klein-Gordon equations.

Upon completion of the course the student must be able to know the topics of the course and know how to derive through the necessary analytical steps the main results discussed in the course. It must also be able to apply this knowledge for the resolution of exercises on the behavior of quantum systems. The aim of the course is also that the student develops the critical capacity for the evaluation of the results obtained. This capacity will be developed during the course, focusing repeatedly on the physical meaning of the formulas obtained and on the methods for evaluating the order of magnitude of the expected results even before carrying out the full calculations.

## Course Structure

Frontal lectures both for the theoretical part of the course (5 CFU- 35 hours) and for the exercises (1 CFU - 15 hours). There will be some exercise classes held as practical tests based on the resolution of the exam exercises of previous years. This activity will be carried on also by a tutor associated to the course. Should the circumstances require online or blended teaching, appropriate modifications to what is hereby stated may be introduced, in order to achieve the main objectives of the course.

## Required Prerequisites

A knowledge of the basic formulation of quantum mechanics is essential up to the resolution of the Schroedinger equation for a one-dimensional square hole, a Coulomb potential and a harmonic oscillator potential. It is important to have basic knowledge of mathematical analysis for the solution of integrals and differential equations and it is useful to know the residual method for the solution of complex function integrals.## Attendance of Lessons

Attendance of the course is usually compulsory (consult the Didactic Regulations of the Course of Studies). The study through advanced quantum mechanics texts alone without directly following the lessons is extremely dispersive and textbooks by their nature are not suitable for helping the student to stimulate the skills of critical evaluation of the results obtained and often do not explain in detail the necessary analytical steps that underlie the main formulas. Furthermore, the reference texts generally contain didactic material much higher than the credits of the course.## Detailed Course Content

**Approximation Methods - **Overview of Time-Independent Perturbation Theory; Interaction (or Dirac’s) representation of quantum mechanics; Time evolution of quantum states: applications to neutrino oscillations; time dependent perturbation theory (instantaneous, periodic, adiabatic); Fermi Golden Rule; Widths of excited states associate to transitions; Applications to the interaction with classical electromagnetic field:photoelectric effect; WKB method and applications to Bohr-Sommerfeld quantization, finite double well potential and tunneling processes.

**Theory of Angular Momemntum and Spin** - Overview of angular momentum and spin eigenstates and communtation relations; Rotations operator; Additions of Angular momenta and spins.

**Foundations of Quantum Mechanics** - Density Matrix formalism, pure and mixture ensembles of quantum states; Einstein-Podolsky-Rosen (EPR) paradox; Einstein's locality principle and Bell's inequality for spin correlation measurements.

**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 for non-relativistic interacting systems; exercises. Response and correlations functions, dynamical susceptibility and spectral representation.

**Primer of Quantum Theory for the electromagnetic field** - Schroedinger equation in a external e.m. field and gauge invariance; Bohm-Ahranov effect and magnetic monopole; simplified approach to the quantization of electromagnetic field; spontaneous radiative emission and dipole transitions.

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

2) F. Schwabl, Advanced Quantum Mechanics, Ed. Springer.

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

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

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

6) B. R. Holstein, Topics in Advanced Quantum Mechanics, Ed. Addison- Wesley.

## Course Planning

Subjects | Text References | |

1 | Teoria perturbativa indipendente dal tempo | 1) and 3) |

2 | Teoria pertubativa dipendente dal tempo | 1) and 3) |

3 | Teoria quantistica dello scattering | 1) |

4 | Formulazione relativistica della meccanica quantistica | 2) and/or 5) |

5 | Scattering in onde parziali | 1) |

6 | Quantizzazione del campo elettromagnetico | 2) and 4) |

7 | Formulazione in path integral | 6) and 1) |

8 | WKB | 3) or 1) |

9 | Density matrix and Bell's inequality | 1) and 3) |

10 | Correlation functions, scusceptibility and spectral representation | 2) |

11 | Addition of Angular Momenta and Spins; rotations matrix | 1) |

## Learning Assessment Procedures

The exam includes both a written test for solving quantum mechanics exercises and an oral test on the different topics of the program.

The written test consists of 2 exercises and lasts 2 and a half hours. The test is considered passed if a score of 18/30 is obtained.

The oral test will cover all the topics covered during the course and generally may also include a comment and any questions on the written test. The criteria adopted for the evaluation are: the relevance of the answers to the questions asked, the level of in-depth analysis of the contents presented, the ability to connect with other topics covered by the program and with topics already acquired in previous years' courses, the ability to report examples, language properties and expository clarity. Verification of learning can also be carried out electronically, should the conditions require it.

## Examples of frequently asked questions and / or exercises

See exercises carried out in class and those already assigned in the last years of the course that are present on the Teams channel of the course, in the "Files" section.

The exercises will mainly be on time dependent and/or independent perturbation theory, on the WKB method, on spin and non-zero angular momentum systems, on scattering theory and on relativistic quantum mechanics.

The questions below are not an exhaustive list but are just a few examples:

- expose the derivation of the time-dependent perturbation theory in terms of the time evolution operator;

- derive the Fermi golden rule;

- describe the Born approximation and discuss the validity regimes;

- discuss the meaning of phase shift in scattering theory;

- derive the Dirac equation;

- discuss and derive the non-relativistic approximation of the Dirac equation in an e.m field;

- outline the fundamental steps in the quantization of the e.m field;

- discuss the difference between mixed and pure state;

- give an example related to Bell's inequality;