Delta Function Scattering with Feynman Diagrams in 1d Quantum Mechanics

TL;DR

This paper demonstrates how Feynman diagrams can be used to explicitly calculate the S-matrix for one-dimensional quantum scattering problems involving delta and finite wall potentials, linking perturbation theory with intuitive conservation laws.

Contribution

It introduces a method to evaluate the S-matrix in 1D quantum mechanics using Feynman diagrams, providing educational insights and a toy model for quantum field theory calculations.

Findings

Explicit S-matrix calculations for delta and finite wall potentials

Illustration of perturbation series summation using Feynman diagrams

Connection between Feynman diagrams and conservation laws in scattering

Abstract

In this paper we will demonstrate the use of Feynman Diagrams for one dimensional scattering in quantum mechanics. We will evaluate the S-Matrix explicitly for the Dirac delta and finite wall potentials by summing the full series of Feynman diagrams, illustrating the spirit of perturbation theory. This technique may be useful in introductory quantum mechanics courses, and provides the student with intuition about conservation laws in the context of scattering problems by connecting Feynman diagrams, free propagation, and conservation of the corresponding observable. It also provides a toy model for calculating S-matrix elements in quantum field theory.