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Canadian Journal of Physics

Path integral solution to a charged particle in an infinite square-well potential with a constant electric field

Kyu Hwang Yeon, Chung‐In Um, Thomas F. George, Lakshmi N. Pandey

Published Sep 1, 1994• 2 citations
Publisher

Abstract

The propagator of a charged particle in an infinite square-well potential in one dimension with a constant electric field is solved exactly using Feynman path integration. The wave function of the system is evaluated by the propagator. This wave function depends on time in a way determined by the well size, particle mass and charge, and applied electric field. The uncertainty relation of the system is also obtained by this function.

DOI
10.1139/p94-075

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The propagator of a charged particle in an infinite square-well potential in one dimension with a constant electric field is solved exactly using Feynman path integration. The wave function of the system is evaluated by the propagator. This wave function depends on time in a way determined by the well size, particle mass and charge, and applied electric field. The uncertainty relation of the system is also obtained by this function.
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.
The propagator for a charged particle in a time‐dependent electric field is calculated following Hida and Streit's framework [1] where the propagator is the T‐transform of Feynman functional. However, we have to regard the potential V = −qE(τ)x≡mℏξ̇x following C. C. Bernido and M. V. Carpio‐Bernido's [2] prescription of time‐dependent potentials. The result agrees with the limiting form of Eqn. (16) of N. Morgenstern Horing, H. L. Cui, and G. Fiorenza [3], when the magnetic field is switched off, and Eqn. (17) of [3] when the electric field is constant in time.