The Effect of the Quantum exp(ipx) in a Potential on a Particle's Probabilities

It is known that a single photon moving in the x direction and encountering an n1-n2 index of refraction discontinuity at xo will either reflect or refract, with the probabilities for each summing to 1. This problem, however, is solved using a balance of Aexp(ipx)+Bexp(-ipx) = Cexp(ip2 x) at xo and the derivative d/dx, of this equation. The complex conjugate of one equation multiplied by the second yields: AA/c = BB/c + CC/c2 where E=pc and E is constant for the incident, reflected and refracted photon. The two continuity equations apply even in the case of a single photon, but in such a case one has either an incident photon changing to a reflected one or an incident to a refracted one. One never has all three together as in thte two continuity equations. We ask: How is it possible to include both a refracted and reflected photon together with the incident in the same equation? 

   We suggest here that the form exp(ipx) applies to momentum, whether it is part of a particle or part of a potential field. exp(ipx) may be obtained from maximization of entropy -P(x)ln(P(x)) subject to the periodic constraint ipxP(x). We suggest that in a single photon interaction, one has momentum in media n1 and n2 and these allow for the continuity equation with the system interactions taking the place of a photon to create the balance. There is stochasticity in that a single photon either becomes a reflected or refracted one, but exp(ipx) does not simply apply to the photon. it may be applied to the n1-n2 interface momentum which is created during the interaction. This interaction is consistent with the a steady stream picture only involving photons

  . Thus for a single photon, it appears as if one has an entangled probability exp(ipx)’s for the incident, reflected and refracted photons, but actually the physical photon only plays a role as first an incident one and then either a reflected or refracted one with the n1-n2 interface momentum creating the missing exp(ipx) piece needed for the continuity balance. Ultimately, A,B and C become probabilities AA/c,= 1, BB/c = P(reflected) and CC/c2= P(refracted), but these are not obtained by maximizing entropy subject to a constraint.as one does not have independence of probabilities.We note that the two continuity equations combine to create a pressure balance which one wishes to have physically in the problem. 

The photon reflection-refraction problem may be reformulated for a particle with rest  mass using a particle with E, p in a constant potential V1 reaching a V1-V2 (both constant) interface at xo with E remaining constant.