On the mean value of the force operator for 1D particles in the step potential

TL;DR

This paper investigates the mean value of the force operator for 1D particles in a step potential within Klein-Fock-Gordon theory, revealing it depends on the discontinuity size rather than the probability density, contrasting with Schrödinger and Dirac theories.

Contribution

It introduces a direct method to calculate the force operator mean value in Klein-Fock-Gordon theory, highlighting a novel dependence on the discontinuity size.

Findings

The mean force is proportional to the size of the probability density discontinuity.

In Klein-Fock-Gordon theory, the force depends on the discontinuity size, not the density value.

Contrasts with Schrödinger and Dirac theories where the force relates to the density at the discontinuity.

Abstract

In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schr\"odinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.