How quadratic potentials fit in

In a recent post, we have discussed how the local-realistic model of quantum mechanics includes external forces. Probability densities of position, momentum, etc., can be derived from those rules of motion, which will be shown in this post. We limit our attention to force fields that standard physics describes as derived from quadratic potentials. In our terminology, the effective momentum transfer ('force' in the standard terminology) generally depends on the lattice node and has the form

For what discussed in the aforementioned post, we treat this force as occurring at each node.

We start considering ensemble of particles emitted by a single source. The general rules of motion prescribe that the span changes its sign at each external reset, that is, by virtue of the "equivalence" assumption above, at each iteration according to the expression

Therefore, there are no possible differences between the span and the trace found. Consequently, external forces alone cannot induce quantum forces.

We can now compute the pdf consequent to external forces. This evaluation is described in the 2017 ArXiv paper that is reproduced here:

For example, for a constant force, f(x) = φ, A = 1, B = t, and C = φt²/2. For a harmonic oscillator, f(x) = -Ωx, A = cos(Ωt), B = sin(Ωt)/Ω, and C = 0.

The general evaluation of probability densities for the 1D particle in the presence of both quantum and external forces will be discussed in a future post.