Quantum exp(ipx) from Classical Considerations, Special Relativity and Discontinuities

We try to argue that the idea of probability associated with motion may have classical beginnings in that the center-of-mass (cm) of a  piece of mass  moving with constant velocity v is present within a dx at some time, but moves out of dx. Flux is constant, but at one time the particle cm leaves dx and arrives at the adjacent dx. Constant flux suggests one does not need to consider time. In such a case, one might associate a periodic function showing probability being removed from a region dx (at x) (i.e. a negative distribution) followed by it being added (a positive distribution) to a region x+dx. This suggests a negative-positive cycle, but this labels certain x regions as negative and others as positive which is inconsistent with all x regions having the same weight based on a constant v. Given that time is removed, it seems this cycling may have nothing to do with velocity v or perhaps anything else for that matter. As a result, one may postulate an Aexp(ibx), where b is a constant, for motion. We will argue for a phase linear in x. AAexp(-ibx)exp(ipx) yields a constant AA, which is flux, and treats all x points the same. 

   At this point, one cannot go farther unless one considers other issues.We consider two such issues. First we ask what happens if the particle is viewed from a frame moving at constant v? The velocity of the particle will not be the same, but we have argued that exp(ibx’) may be the form, yielding no information about b when (x,t) changes to (x’,t’). If on the other hand, the phase cannot change if the particle is viewed from the moving frame, one requires a Lorentz invariant. This suggests that the phase may actually be -Et+px. (We also show that it is not necessary to use special relativity to obtain the form exp(ipx). Rather one may consider ideas of pressure balance associated with continuity of exp(ibx).)

  As a second consideration, we examine a one dimensional reflection-refraction problem at a point xo for which a  probability equation 1= Probability(reflect)+Probability(refract) exists. If exp(ipx) for constant E describes motion,  this motion changes with changing p, yet one wishes probability to be a continuous function at xo. This suggests different weight coefficients for different exp(ipx) which seem to be associated with P(reflect), and P(refract). If x=0,  one has an equation in the weights, with A+B (x<xo) matching C (x>xo) at xo. If one stops at this point,  exp(ipx) does not seem to have any impact. One may note, however, that in general xo is not zero and that one wishes at the very least to have the derivatives match. This immediately links the A,B,C values to the p (momentum) discontinuity. Thus using special relativity to create an exp(ipx) has practical implications because it allows one to deal with p discontinuities. 

     We also show though that special relativity, i.e the use of -Et+px, may be bypassed and the form Aexp(ipx) obtained as a result of its first derivative yielding p which is needed in a pressure balance equation. This then suggests A is the square root of flux.