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<n2. As a result the negative value of B allows for conservation of probability (or photons). At first this may seem a mathematical convenience, but the two equations ((1a)) and ((1b)) lead to physical consequences, namely the probabilities BB and CC in terms of AA. We thus speculate that there might be a physical mechanism associated with the minus value of B, the reflected ray. This physical property (which should exist for individual photons)  ensures conservation of probability. 

  If one considers a glass wedge (or thin film) in air one has reflection from the glass (n1<n2) bringing in a minus sign in B (for conservation with the incident beam). The refracted beam becomes the new incident beam within the glass and reflects at an n1>n2 interface so B’ is not negative. B and B’ may now interfere  (depending on the thickness of the wedge) creating an interference pattern which must conserve probability overall. In general one would expect B and B’ to be independent, but each carries conservation of probability information (in a physical form) and this information is different causing interference to occur.