Information, Quantum Probability and Reflection/Refraction

 Quantum mechanics is known for its “square root” probability i.e. the existence of a wavefunction W(x) such that P(x) = W*(x)W(x). This begs the question: Why is this the case? In Maxwell’s electromagnetic theory of the photon, a similar situation arises except that one has energy density instead of P(x) which consists of .5C1 Electric field* Electric field + .5c2 Magnetic field*Magnetic field. The  “square root” object is a physical observable (Electric field, Magnetic field) which both behave as cos(-Et+px) and so, more accessible than W(x).

    Alternatively one may use Fermat’s least time principle for light to show that i two dimensional problems (reflection/refraction)  d/dx (used in extremization) is really the operator of momentum in that direction i.e. a spatial invariant direction. Thus A=px such that dA/dx=p = x component of momentum which may be generalized to A=-Et+p dot r.  (p.r) may be written as -i ln(exp(ip dot r) with exp(i p dot r) representing a “kind of probability”. This probability may be considered a square root probability because for uniform motion (light) all x points along a ray should have equal probability. Thus exp(i p dot r) maps into a constant which happens if one takes the modulus.

  Both of the above approaches (using continuity at x or a kind of conservation of probability and a momentum) may be used to solve the problem of an incident beam of light moving in the positive x direction hitting a medium and both reflecting and refracting. For example see (1). The goal is to find the two intensity values associated with the reflected and refracted beams assuming the incident intensity is known. 

   In this note, we take a different approach to finding the two unknowns, one that does not use the idea of the electric field proportional to cos(-Et+px) or the probability exp(ipx). We wish to solve the problem using only the idea of information and stress the importance of beginning with energy conservation a priori unlike the two approaches listed above.

   We postulate that there are two sets of information in this problem. First there is the notion of Energy = p(momentum in a vacuum) n c/n where n is the index of refraction. Thus the incident and reflected light are associated with pc1,c1 and the refracted with pc2, c2 with energy being the same for all. Second there is the idea of conservation of energy i.e. if ten photons with energy E hit the surface and a10 reflect and (1-a)10 refract, then the sum of the refracted and reflected energy equals the initial energy. Energy is then equated with intensity through Intensity/c = energy (1).

   Using only these two pieces of information we argue one should be able to find the intensity of both the reflected and refracted ray. This leads to an immediate problem because there are two unknowns and only one equation, namely conservation of energy i.e.  I(A)/c1 = I(B)/c1 + I(C)/c2 where A is the incident beam, B the reflected and C the refracted. Writing as:  I(A)/c1 - I(B)/c1 = I(C)/c2  (i.e. a spatial instead of time grouping) suggests  a possible solution might be to employ a difference of squares noting that 1/c = p/E with E being a constant in the problem. Thus one may reduce the problem to A+B=C and p1A-p1B=p2C. This is of the form of a new kind of “square root” type probability, but leads to two equations with two unknowns (A is assumed to be known) which allows the problem to be solved.