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

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