Project III 2024-25
Path Integrals and Instantons in Quantum Mechanics
Madalena Lemos (email)
Description
Classically if you have a particle in a double well potential as the one pictured on the right there is no trajectory for a particle to travel from one well to the other. Quantum mechanically, however, a particle can tunnel from one to the other -- these tunneling processes are described by Instantons .
To study instantons we will first study path integrals. In quantum mechanics you quantized by promoting variables to operators, this project explores an alternate quantization approach, called path integral quantization. We will quantize the theory by summing over all possible trajectories, or more precisely by a weighted integral over all possible trajectories, not rescricted to the path that is the solution of the classical equations of motion. Motivating this approach with the double slit experiment (recall the electron "sees" both slits) you will derive the path integral, and see how in the classical limit you recover the principle of least action principle of least action. Path integrals are a beautiful way to describe Instantons . These appear naturally as classical solutions to the Euclidean equations of motion with a finite action, so you can study of instantons in quantum mechanics.
There are many uses of path integrals in quantum mechanics that could also naturally fit into this project.
Essential Prior Modules
- Mathematical Physics II or Theoretical Physics 2
Essential Companion Modules
- Quantum Mechanics III or Foundations of Physics 3a
References
- Feynman and Hibbs, Quantum Mechanics and Path Integrals -- chapters 1, 2
- J. Zinn-Justin, Path integrals in quantum mechanics
- T. Lancaster, S. Blundell, Quantum Field Theory for the Gifted Amateur -- chapters 23, 50
- Coleman, Aspects of symmetry chapter 7.2