Impulse versus Pressure or Quantum Versus Classical in Reflection / Refraction

In classical statistical mechanics, e.g. a Maxwell-Boltzman (MB) gas, one calculates pressure by multiplying 2p (impulse) by v (velocity or flux). This is done because one has multiple particles and so flux is important. One needs to know how many particles strike a surface per sec as well as their impulse. What happens if one has a single photon (or particle)? This is not a steady stream situation, although one may create a corresponding steady stream situation. In particular, a single photon moving in the positive x direction may reach an interface of two media with different indices of refraction and either reflect or refract. In such a case one has a probability to reflect and one to refract which add to 1. Thus there are two unknowns (i.e. the two probabilities) and energy stays constant. 

   One may immediately write a conservation of energy equation and apply it to a steady state scenario. This necessarily utilizes the idea of velocity because there is a flux. This energy conservation equation may be rearranged to give the appearance of a pressure balance equation which also depends on velocity i.e. flux. The problem is that conservation of energy or pressure balance is a single equation and there are two unknowns. One may not simply say that the probability to reflect is  .5 and that to refract .5.

     In fact, in a previous note (1) we solved this problem using the idea of writing the energy equation in terms of two equations linked to differences of squares. We suggested that these two equations which only involve weights are linked to a function which together with its first derivative is continuous at x=0 .In (2) we suggested that the x-distributions cos(px) and sin(px) represent action-reaction impulse type distributions. We argued that an impulse creates momentum p in the first place and that this “impulse”, in the form of x-distributions, is still present because the object may strike a foil and impart impulse at any time. Furthermore, because cos(px) and sin(px) are probability distributions, there is no sense of time within a wavelength. On average, however, the object moves with a constant velocity. In the case of light it is c/n where n is the index of refraction and in the case of a nonrelativistic particle p/mo. 

    The example of a single photon either reflecting or refracting allows one to formulate the problem in terms of impulse i.e in terms of the probability distribution scheme such that one does not have time and hence velocity. This scheme requires two equations. This impulse based x-probability distribution scheme or quantum mechanical approach, however, should map into the classical pressure balance because on average a photon does move as x=c/n t which it does.  It is, however, the quantum approach (impulse picture with no time or velocity within a wavelength) which solves the problem i.e. yields the probability to reflect or refract which may then be linked to classical type (steady stream intensities and flux.