Information, Quantum Probability and Reflection/Refraction Part III Interference

Note Dec. 8, 2022: B in A+B=C and p1A-p1B=p2C is negative for n1<n2 where n is the index of refraction. Both equations i.e. probability conservation and average momentum conservation are involved in this result i.e. B/A = (c2-c1)/(c1+c2) (1). In Parts I and II we considered the problem of a beam of light A moving in the positive x direction striking a medium at x=0 and reflecting B and refracting C. This is a probabilistic problem because a single photon has a probability to reflect and another to refract. We also noted that conservation of energy holds i.e. I(A)/c1 = I(B)/c1 + I(C)/c2 where I(A),I(B) and I(C) >0 are intensities . The goal is to find I(B) and I(C) given I(A), but there is only one equation and two unknowns. In Part I we argued that an energy conservation equation incorporates two equations representing square root type probabilities. In Part II we argued that a physical motivation for such a situation may be that in a steady state case, one has dynamic directional probabilities not just static ones. The static/classical probability is associated with the conservation of energy and groups by time i.e an incident beam exists during one time range and reflected and refracted beams during another. One, however, may create a steady state system so that all three beams exist at the same time, although the reflected and refracted beams at any instant are the result of incident photons which hit the medium at an earlier time. A steady state system should not rely on time, but still contains probabilities. There is thus no reason to group by time, but rather one groups by space with the incident and reflected beams in one region and the refracted in another. Introducing space grouped dynamic probabilities A,B,C one may write (using conservation of probability and average momentum): A+B = C ((1a)) and p1A -p1B = p2C ((1b)) where p1=n1p and p2=n2p and c1=c/n 1 and c2=c/n2. The second equation has the appearance of a conservation of energy equation if one places the term -p1B on the RHS. The problem is that A+B=C implies B must be negative if n1<n2. Thus dynamic probabilities may be negative (which is unusual) and the second equation cannot be conservation of energy although it is of the correct form. In Part II we noted one may multiply the two equations together and by using a difference of squares preserve the form of an energy conservation equation which now has AA, BB and CC terms which are necessarily positive and may be identified with I(A),I(B) and I(C). In this note we wish to consider the consequences of B being negative. This we suggest implies A and B interfere so that A+B yields a number smaller than A i.e. C for n1