ADVANCED QUANTUM MECHANICS
Academic Year 2024/2025 - Teacher: VINCENZO GRECOExpected 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
Attendance of Lessons
Detailed Course Content
Approximation Methods - Overview of Time-Independent Perturbation Theory focusing degenerate state case; 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 states in quantum 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 Momentum and Spin - Overview of angular momentum and spin eigenstates and commutation relations; Rotations operator on spinors.
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.
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; Casimir effect.
Path-Integrals - Propagators and Green-functions; Path-Integral formulation of quantum mechanics; Examples: free particle, harmonic oscillator; primer on instantons.
Relativistic Quantum Mechanics - Klein-Gordon Equation and Klein’s paradox; Dirac Equation and the free particle and anti-paticle solutions; Weyl and Majorana representations; Non-relativistic reduction of Dirac equation: Pauli equation; Charge and Parity simmetries; Dirar equation coupled e.m. field: first order relativistic corrections; hyperfine structure and Lamb-shift; exercises.
Textbook Information
1) J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, Ed. Addison-Wesley./Meccanica Quantistica Moderna, Ed. Zanichelli (III edizione)
2) F. Schwabl, Advanced Quantum Mechanics, Ed. Springer.
3) Giuseppe Nardulli - Meccanica quantistica: applicazioni, vol II, Ed. Franco Angeli.
4) J.D. Griffiths, Meccanica Quantistica, Ed. CEA (II edizione)
5) J.J. Sakurai, Advanced Quantum Mechanics, Ed. Addison-Wesley.
6) B. R. Holstein, Topics in Advanced Quantum Mechanics, Ed. Addison- Wesley.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Teoria perturbativa indipendente dal tempo | 1) 3) or 4) |
| 2 | Teoria pertubativa dipendente dal tempo | 1) 3) or 4) |
| 3 | Teoria quantistica dello scattering | 1) and 4) |
| 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 | 1) 3) and 4) |
| 9 | Density matrix and Bell's inequality | 1) and 3) |
| 10 | Casimir Effect | 1) e 6) |
Learning Assessment
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;
- derive the adiabatic approximation and the Berry phase;
- 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;