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Journal of Physics A Mathematical and General

Explicit derivation of the propagator for a Dirac delta potential

Edward B. Manoukian

Published Jan 7, 1989• 38 citations
Publisher

Abstract

An explicit derivation of the propagator is given for a particle in a Dirac delta potential lambda delta (x) by making a direct application of a variational technique followed by a systematic use of Laplace transforms. Although the solution of the corresponding Schrodinger equation has been known for years, the solution of the propagator has been obtained only recently.

DOI
10.1088/0305-4470/22/1/013

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An explicit derivation of the propagator is given for a particle in a Dirac delta potential lambda delta (x) by making a direct application of a variational technique followed by a systematic use of Laplace transforms. Although the solution of the corresponding Schrodinger equation has been known for years, the solution of the propagator has been obtained only recently.
Abstract We obtain accurate eigenvalues of the one-dimensional Schrödinger equation with a Hamiltonian of the form H g = H + gδ(x), where δ(x) is the Dirac delta function. We show that the well known Rayleigh–Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh–Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.
The authors, together with A. del Campo, developed an alternative method for determining the Feynman propagator [1] for a non-relativistic problem. One started with the time dependent Schroedinger equation for the problem. Carried out a Laplace transform with respect to time to get the equation for the energy dependent Green function and derived it explicitly. We then carried out the inverse Laplace transform in the energy to get Feynman propagator. In this paper we carry out the same programme for a relativistic problem associated with the one dimensional Dirac equation of a free particle and the Dirac oscillator proposed by Moshinsky and Szczepaniak [2] twenty years ago.
Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
The solution of Dirac particles confined in a one-dimensional finite square well potential is solved by using the path-integral formalism for Dirac equation. The propagator of the Dirac equation in case of the bounded Dirac particles is obtained by evaluating an appropriate path integral, directly constructed from the Dirac equation. The limit of integration techniques for evaluating path integral is only valid for the piecewise constant potential. Finally, the Dirac propagator is expressed in terms of standard special functions.