# Quantum physics IV

## 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. Semiclassical 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, II and III

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 a 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 in lectures. Solving problem sets during exercise hours. Critical study of the material.

## Assessment methods

Written exam

## Supervision

Office hours | Yes |

Assistants | Yes |

Forum | Yes |

## 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.

## Ressources en bibliothèque

- Quantum Mechanics and Path Integrals
- Modern Quantum Mechanics
- Path Integrals in Quantum Mechanics, Statistics and Polymer Physics
- Path Integral Methods and Applications
- Techniques and applications of path integration
- Aspects of Symmetry

## Notes/Handbook

Prof R. Rattazzi: Lecture Notes for Quantum Mechanics IV

## In the programs

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Quantum physics IV**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Quantum physics IV**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Quantum physics IV**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks

**Semester:**Spring**Exam form:**Written (summer session)**Subject examined:**Quantum physics IV**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks