PHYS-426 / 6 crédits

Enseignant: Augusto Penedones Fernandes João Miguel

Langue: Anglais

## 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.

## 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.

Written exam

## Supervision

 Office hours Yes Assistants Yes Forum Yes

## 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

Lecture Notes for Quantum Mechanics IV

## Dans les plans d'études

• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Quantum physics IV
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Quantum physics IV
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Quantum physics IV
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Printemps
• Forme de l'examen: Ecrit (session d'été)
• Matière examinée: Quantum physics IV
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

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