PHYS-426 / 6 credits

Teacher(s): Carleo Giuseppe, Rossi Riccardo

Language: 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.
• 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
• 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 and II

## 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.

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

"Path Integrals in Quantum Mechanics", Jean Zinn-Justin, Oxford Graduate Texts, 2010.

## 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional
• 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
• Type: optional

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