Speculation on Quantum Free Particle Probability and Spatial Equilibrium

   We argue that the toss of a coin or a die may be seen as an equilibrium in time. In particular, for a coin there is an intrinsic spatial feature (two sides of the coin) which a priori defines probability for heads/tails and after a very large number of tosses, one essentially has 50% of the results as heads and 50% as tails. In such a case, time runs in one direction. One may run a movie backwards, but then time still runs in one direction, i.e. backwards. Spatial equilibrium, however, involves motion in space in both directions, as in the case of pressure in an ideal gas with no potential V(x). Thus, we argue that for a spatial equilibrium, one must consider both directions of space. This leads to an interesting situation if one has a directional problem associated with a spatial equilibrium equation which should not be directional based.

   In particular, we consider the example of one dimensional reflection-refraction at an n1-n2 index of refraction junction. At first this has the appearance of an equilibrium in time through: 1= P(reflect) + P(refract). Just like the coin toss, after many photons N impinge at the junction, there will be N P(reflect) which reflect and N P(refract) which refract. In a previous note, we saw that one may write this probability conservation equation in terms of dynamic variables, i.e.

AA/c - BB/c = CC/c2 —>   AAp - BBp = CC p2 ((1)), using E=pc with E being the same for the incident, reflected and refracted photons. Here AA,BB,CC are the fluxes of the three.  We also noted that written in this way, one has pressure balance. We argue here that a pressure balance equation represents a spatial equilibrium and only makes sense in terms of motion in both x directions. 

  A steady stream case or a single photon scenario, however, only involves one direction of motion. We thus argue that one must introduce equations which explicitly show directional motion and that these must create the pressure equilibrium equation which has not sense of direction. To do so, we consider AA, the incident flux and note that flux is the number of particle per sec. As a number it has no direction, although one may associate it with a velocity vector. We suggest that for a pressure spatial equilibrium case, it must be associated with both directions of motion (i.e. x) and that one should write a probability in terms of p which governs interactions. This suggests:   AA = Aexp(ipC) Aexp(-ipC). (Here C is a constant to account for units.) In other words, if one performs a probabilistic equilibrium calculation one must have a proper equilibrium with motion to the left and right described. A physical interaction of a single photon, however, only involves one direction and so is linked with one piece  A exp(ip) we argue. Furthermore, as a probability this should be continuous in x and so instead of exp(ipC), suggest A exp(ipx) as the proper probability. We then see that changing the direction of the x-axis leaves exp(ipx) unchanged. As a result, one tries to establish an equation which involves the incident photon moving to the right or left, but this must ultimately be associated with a pressure equilibrium equation ((1)) which must involve both directions of the incident flow, i.e. an equation in exp(ipx) and its complex conjugate. We suggest that this is the reason that a so-called square root Aexp(ipx) of flux emerges from AA, i.e. free particle quantum mechanical behaviour.