Quantum physics IV

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

I. The Path Integral Approach to Quantum Mechanics

  1. The Feynman Path Integral in Quantum Mechanics
  2. Thermal Density Matrix and Imaginary-time Path Integral
  3. Propagator of the Quantum Harmonic Oscillator

II. Interaction with an external electromagnetic field

  1. Gauge invariance in Quantum Mechanics
  2. Particle in a constant magnetic field and Landau levels
  3. Path integral for a particle in a vector potential and Aharonov-Bohm effect. Introduction to the geometric phase

III. Correlation functions, functional methods, and Feynman diagrams

  1. Path integral for the matrix elements of operators, path roughness and thermal correlation functions
  2. Functional methods: Wick's theorem, Feynman-Jensen variational principle and mean-field theory
  3. Feynman diagrams: Wick's theorem, linked-cluster theorem, and application to the quartic anharmonic oscillator

IV. The Semiclassical Approximation and Instantons

  1. The semiclassical spectrum
  2. The semiclassical Van Vleck-Pauli-Morette propagator
  3. Instantons in Quantum Mechanics: the gap of the quartic double well

Keywords

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

Learning Prerequisites

Required courses

Analytical Mechanics

Physics III (Electromagnetism)

Quantum Physics I, II

 

Recommended courses

Quantum Physics III

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.

"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.

Ressources en bibliothèque

Notes/Handbook

 

 

Moodle Link

In the programs

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

Reference week

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