Joint Phase Space (x-p) Probability and Free Particle Quantum Mechanics

   There seems to be much interest in the literature (1) in associating quantum mechanics (in particular the Schrodinger equation) with a classical joint phase space probability F(x,p,t) such as appears in classical Liouville theory etc.

   Here, we consider that P(x)dx = dx/L for a particle at rest and analyze the extra physics which appears when one considers a single particle bouncing back and forth in a box, which is what a person in a frame moving with -v would see (albeit with a Lorentz contracted L.)

  We suggest that the constraint L is important as it is linked with physical forces which appear when the particle moves. In other words, a particle at rest is linked with a static probability, while for motion, one must have a probability which is linked to more than x and t, but to force considerations in x and t, a dynamical probability. (Experiment bears this out as we discuss.)

 In particular, we argue that in the case of a moving particle, two physical concepts related to force appear, namely pressure which is an average over time and impulse hit which is only linked with space. Thus, two different particles may have the same p, but different v and pressure values. We thus suggest a physical grouping of (p,x) and (pressure, t). We then note that one needs to somehow preserve the notion of P(x)=1/L, i.e motion does not introduce a bias to x, nor to time. 

 In a rest frame, there is no motion and so there is neither p nor pressure, but P(x)=1/L pertains. If one tries to define a probability with two groupings, one for pressure, t and the other for p,x, then if one multiples a p1 case by a p2 case, the p,x grouping should yield p1+p2, x. Furthermore, a p,x grouping should imply mixed terms, e.g. px etc, otherwise one does not really have an association between the two variables. This suggests a pf(x) form linked to the p-x grouping. Given the notion of special relativity, it seems that one may replace pressure with the notion of energy and create a Lorentz invariant probability. This would then lead to the free particle quantum wavefunction (probability) exp(-iEt+ipx), but we argue that the roots are in classical physics and a desire to physically describe the physical information present for a particle bouncing back and forth in a box.