Reflection/Refraction, Spatial Invariance and Origin of Statistical Nature of Problem

It is known that if a photon hits a surface with a different index of refraction (at an angle within a certain range), it has a specific probability “a” to reflect and 1-a to refract. In this note we ask: Where does this statistical nature come from?

   In previous notes, we have considered Fermat’s principle applied separately to explain two dimensional reflection and refraction. We have also used the idea of a difference of squares and energy conservation in one dimensional reflection/refraction to find values of the probabilities. The question we ask is: Why are there probabilities in the first place? Is there some general idea at work despite the complicated details of the photon interacting with the atoms of the new medium.We argue the general notion  is due to the idea of spatial invariance which is linked to the translation operator d/dx which in turn in linked to a p impulse occurring at a given x .Thus the variables p and x are linked through the notion of an impulse existing at some x.  The idea that d/dx may operate on a function of constant p and x to yield p, implies the function of p,x should have physical meaning. This is a problem as classically a particle moving with constant p should not be associated with any function of x except a constant. (All x points are the “same”.) 

   Thus we argue that a “probability” in x is established with a constantly moving particle (or even photon). The eigenfunction of d/dx (or rather -id/dx) which exhibits some kind of spatial invariance is exp(ipx). This contains two different x probability distributions cos(px) and sin(px). We noted that d/dx is associated with p due to  an impulse hit at say an x point. Thus various exp(ipx)s with different weights should appear together in a physical problem with impulses such as reflection/refraction. The weights of the various exp(ip1 x), exp(ip2 x) etc become linked to probabilities associated with the state 1, 2 etc. Thus the origin of these probabilities (i.e. their specific values) follow from the probabilistic form exp(ipx) which follows from d/dx. (In (1) we have argued that the x distributions in exp(ipx) may be further linked to the force which “created” the p in the first place, with different p values having different distributions cos(px), sin(px)).