Quantum Free Particle exp(ipx) With p Being Dynamic and x Static Part II

In Part I, we noted that the classical probability expression P(x)dx= Cdx/v(x), where v(x) is velocity from .5mo v(x)v(x)+ V(x)= E (1), sometimes appears in the literature. For v(x)=constant, P(x)dx=dx/L, where L is length.  This probability is based on the amount of time, dt, a particle spends in each dx. It could, however, just as well be the probability of finding a rest mass somewhere in L, i.e. there are no dynamics in P(x), we argue.

    The use of probability in classical mechanics is interesting because it is deterministic and should not have probability if one follows a particle with constant v through x=vt. Thus, as a first step, P(x)=1/L seems to mean that one is not following the particle in time so it may be anywhere in L and have a direction either to the right or left. 

   In Part I and previous notes, we have suggested a more consequential condition, namely that P(x)=1/L means that one has Pd(p,x)Pd(-p,x) at each x for this to be equivalent to having a rest mass anywhere in L. In other words, one has a product probability for p and -p at each x. A particle, however, has either p or -p, it cannot have both at the same time, but one does not follow the particle in time for P(x). By breaking P(x) into two factors, the sense of motion and direction appears, but a paradox seems to arise. One must have p at each x point, but at the same time probability is disappearing at x for p and -p because momentum necessarily implies motion. A way to demonstrate decreasing probability together with a  p at each x is to have a two dimensional probability such that the magnitude is 1, but each component shows a change in Pd(p,x) which we call a dynamical probability. If one dimension decreases in probability, the other increases etc in a periodic manner. This leads to Pd(p,x)=exp(ipx) as a possible solution. This solution is unusual and does not seem to match classical observation, but it does explain two-slit interference which is also experimental as well as one dimensional reflection-refraction.

   We suggest that if one uses a purely dynamical motion, x=vt, then there is no probability. The view of probability appears through the notion of p (and -p)  being at each x at the same time that probability (in each dimension) changes at x, but in a different way for p and -p.

    One may note that in real life problems, one has motion in one direction, e.g. interference from 2-slits or one dimensional reflection-refraction. These problems, however, cannot be solved using Newtonian mechanics. If one tries a probabilistic approach, we argue that it must be substantially different from the x=vt view. We suggest that taking time out of the picture means that one must have p at each x and yet account for probability flow in one direction or the other. This leads to the concept of a wavelength and periodic probability. In such a case, x in exp(ipx) must be considered static.