Quantum Dynamical Probability Wave Part II (High Energy Limit)

In part I we argued that classical mechanics divides a one dimensional line into equally spaced dx regions. These are small, but not zero because classical spatial density is C1 dt = C1 dx/ v(x) where v(x) is velocity which is assumed to be constant within dx to first order. In terms of probability, if a particle is within a certain dx, it is not in the adjacent ones i.e a probability distribution reaches zero at the endpoints of dx. We argued that for a constant velocity classical density is P(x)=constant and so one must keep two spatial probability distributions (in two dimensions or separated by i) such that the modulus is 1. This makes the two ideas compatible.

    This idea seems to apply to quantum mechanics with the stipulation that dx is linked to 1/p. Thus one has exp(ipx) as a dynamical probability as discussed in Part I. In this note we wish to consider this in more detail together with the Schrodinger equation. In particular we focus on the high energy limit.  We argue that exp(ipx) should hold for a small dx region classically which may include several wavelengths proportional to 1/p. Thus there is still the idea of the distribution and its shift as two dimensions, but we argue that the time-independent Schrodinger equation may be an approximation as it considers exp(ipx) and exp(-ipx) when creating Wn(x) and it is not clear that such interference holds in a very high n limit.