Feynman propagator for the δ-function potential

Ilaria Cacciari, Paolo Moretti

S. M. Blinder

A particle in a one-dimensional \ensuremath{\delta}-function potential possesses both discrete and continuum solutions. The configuration-space Green's function and propagator for this problem are derived by explicit summation over the spectrum of eigenstates. The momentum-space Green's function is also obtained. The propagator does not contain the classical action function in any simple way, in contrast to the usual structure in Feynman's path-integral formalism. Various analogies between the \ensuremath{\delta}-function and Coulomb problems are discussed.

V.I. Karloukovski

James P. Edwards, Víctor A. González-Domínguez, Idrish Huet +1 more

We outline an approach to calculating the quantum mechanical propagator in the presence of geometrically nontrivial Dirichlet boundary conditions. The method is based on a generalization of an integral transform of the propagator studied in previous work (the so-called "hit function") and a convergent sequence of Padé approximants that exposes the limit of perfectly reflecting boundaries. In this paper the generalized hit function is defined as a many-point propagator, and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in D=1 and D=3 dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, nonlocalized potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.

L. D. Salem, Horacio S. Wio

Zakariah Crane

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.