# Fiches de cours

## Quantum physics IV

Vacat .

English

#### Summary

Introduction to the path integral formulation of quantum mechanics. Derivation of the perturbation expansion of Green's functions in terms of Feynman diagrams. Several applications will be presented, including non-perturbative effects, such as tunneling and instantons.

#### Content

1. Path Integral formalism

• Introduction
• Propagators and Green's functions.
• Fluctuation determinants.
• Quantum mechanics in imaginary time and statistical mechanics.

2. Perturbation theory

• Green's functions: definition and general properties
• Functional methods
• perturbation theory by Feynman diagrams

3. Seminclassical approximation

• The semiclassical limit

4. Non perturbative effects

• reflection and tunneling through a barrier
• Instantons

5. Interaction with external magnetic field

• gauge invariance in quantum mechanics
• Landau levels.
• Aharonov-Bohm effect.
• Dirac's magnetic monopole and charge quantization.

#### Keywords

Path integral formalism. Green's function. Determinants. Feynman diagram. Feynman rules. Perturbation theory. Non-perturbative effects. Tunnelling. Instantons. Gauge-invariance.

#### Learning Prerequisites

##### Recommended courses

Quantum physics I and II

Quantum Field Theory I

##### Important concepts to start the course

Solid knowledge and practice of calculus (complex variable) and linear algebra

#### Learning Outcomes

By the end of the course, the student must be able to:
• Formulate a quantum mechanical problem in terms of a Path integral
• Compute gaussian path integral as determinants
• Express physical quantities in terms of the Green function
• Translate a Feynman diagram into a mathematical expression
• Compute a Feynman diagram
• Compute tunneling rates in simple quantum potentials
• Formulate the quantum theory of an particle interacting with an external electromagnetic field

#### Transversal skills

• Use a work methodology appropriate to the task.
• Set objectives and design an action plan to reach those objectives.

#### Teaching methods

Ex cathedra and exercises

#### Expected student activities

Participation to classes. Solving problem sets during exercise hours.

Oral final exam

#### Supervision

 Office hours Yes Assistants Yes Forum No Others Office hours: Wednesday 14-15

#### Resources

##### Bibliography

"Quantum Mechanics and Path Integrals" , R.P. Feynman and A.R. Hibbs, McGraw-Hill, 1965.

"Techniques and applications of Path Integration',  L.S. Schulman, John Wiley & Sons Inc., 1981.

"Path Integral Methods and Applications", R. MacKenzie, arXiv:quant-ph/0004090.

"Modern Quantum Mechanics',  J.J. Sakurai, The Benjamin/Cummings Publishing Company, 1985.

"Aspects of Symmetry", S.  Coleman, Cambridge University Press, 1985.

"Path Integrals in Quantum Mechanics, Statistics and Polymer Physics', Hagen Kleinert, World Scientific, 1995.

##### Notes/Handbook

Prof R. Rattazzi: Lecture Notes for Quantum Mechanics IV

http://itp.epfl.ch/webdav/site/itp/users/174685/private/RevisedLectureNotesV2.pdf

### Semaine de référence

LuMaMeJeVe
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
En construction

Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand