Quantum Features from Time-Independence of Impulse Balance in the Photon Reflection/Refraction Problem

    If one considers the reflection/refraction of a steady stream of photons moving in the positive x direction and striking an n1=1 versus n2 interface along the y axis at x=0,  one may immediately write a conservation of photon number equation i.e. N(incoming) = N(reflected) + N(refracted) ((1)). This equation, however, does not shed any light on the ratio of N(reflected) to N(refracted) for a given N(incoming). One may also notice that no information about the index of refraction is included. This information manifests itself in:  E= p1 c/n1 = p2 c/n2. 

    Taking ((1)) and multiplying by e (photon energy) yields a conservation of energy equation. In both the cases of photon number and energy conservation there is no notion of flux. Taking the energy conservation equation and writing e=p1c1 or e=p2c2 yields an equation relating momenta (proportional to impulses), but this equation involves dynamics i.e. fluxes i.e. c1, c2 values. Thus it is not on the same footing as the photon number or energy conservation equations. 

    We argue that there should exist an impulse balance type equation which does not depend on flux. We suggest this is a key physical idea. (For example, the two-slit interference pattern of p=m1v1=m2v2 are the same.) As a result, the single equation ((1)) is not sufficient for solving the problem. To solve the problem, one needs to recast ((1)) in terms of flux (a notion which exists classically) divided by ci (speed of light) so that one obtains the number of photons. By having 1/c1 appear in the equation, one may multiply by e so that e/ci = pi. 

    One then has the equation:   AA/c1 = BB/c1 + CC/c2 ((2)) where AA=incoming flux, BB=reflected flux and CC refracted flux. One must change from a static picture of counting photons to a flux picture, but divide fluxes by ci’s to regain the static picture. Incoming flux is written as AA to indicate it is positive etc. ((2)) still does not solve for B and C in terms of A and n1,n2, but one may bring BB/c1 to the LHS and break ((2)) into two equations using differences of squares. Thus one has an A+B=C equation which is like a nonflux balance of probability or energy (if one multiplies by e) and a static momentum/impulse balance equation pA-pB= p2 C.

Solving the problem involves creating not only time independent photon number/energy balance, but also time-independent photon momentum/impulse balance. 

    We argue that the breaking of ((2)) into two equations introduces the notion of quantum mechanics because one uses “square roots” of flux probability. One may ask why one does not simply find a second equation in addition to the photon number/energy balance equation. We argue that the single number balance equation contains all of the necessary information in the problem i.e. flux, photon number, energy or p’s and c’s. Thus a second equation would not be independent. As a result, one must break the single photon balance equation into two independent equations leading to a different kind of physical balance i.e. one involving square roots of fluxes instead of fluxes i.e. quantum mechanics.