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

 In Part I, we noted that a photon number balance equation , which is equivalent to an energy balance equation, written in terms of fluxes describes all of the information in a photon steady state reflection/refraction problem in one dimension. We also pointed out that one equation cannot solve for two unknowns and suggested that because all information is contained in the classical photon count equation, one may break it into two equations (quantum balance equations for probability and average impulse) in order to solve the problem. Thus the two quantum equations are needed to solve the problem, but collapse into the classical conservation of photon number equation.

   In this note, we argue that in classical mechanics there are two different, but consistent descriptions of force acting on a particle. The first is impulse Integral dt F = delta p  where p is momentum. Two particles with the same momentum, but different velocities, create the same physical impulse. In other words, time is removed (integrated out) in this picture.

   An equivalent formalism is to write dt = dx/v or   F dx = v dp.  This form breaks the degeneracy of p’s with different velocities. In other words, time is important.

  Thus we argue that at the classical level there are two equivalent formalisms. The impulse one does not consider time, while the work-kinetic energy one does. Both are legitimate.

    For the reflection-refraction of photons in one dimension, both formalisms should apply and lead to consistent results as do the impulse and kinetic energy-work approaches. The latter uses time and is equivalent to three ideas: photon number conservation, energy conservation and pressure conservation, all three being associated with time. The concept of impulse, however, suggests that one should be able to consider space in terms of impulse alone and not time i.e. not velocity. This forces one to introduce a complex probability associated with impulse hits i.e. exp(ipx). This probability and its corresponding average impulse balance yield two independent equations, while the photon count conservation only supplies one. Thus the problem needs to be solved using impulse formalism, even though it is completely consistent with count conservation.

    The form exp(ipx) leads to new physics (when compared with classical physics) as it suggests interference pattern results.