Published August 15, 2024 | Version v1
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Conservation of Momentum and a Photon-Only View in 1D Reflection-Refraction

Description

A classical free particle whose energy is described by EE = pp + momo (c=1) or E=pp/2mo has two degenerate solutions in terms of momentum information, namely p and –p. These are time reversed situations of each other. In order to realize both in a problem, one requires reflection from a surface, i.e. a force and overall conservation of momentum must exist. Given that the particle energy, however, does not change, if one only considers energy as information one might argue that nothing happens when p becomes -p. In other words, p and -p may be seen from the point of view of degenerate solutions linked to the same E in a particle-only view.

    In the case of a photon reflecting and refracting, E is again one value. Thus, an incident, reflected and refracted photon may be considered different degenerate solutions in the information momentum. As argued in previous notes, given that E is the same, nothing happens if one only considers E. In terms of p, however, impulses occur. In this case, an incident photon either becomes a reflected one or a refracted one and there is a probability. This probability, however, only appears if one considers the incident, reflected and refracted photons together in one equation even though physically one has an incident photon becoming a reflected or a refracted at different times. Thus, an equation (or two because there are two unknowns) seem to be somewhat mathematical because different times are mixed which is even what happens in classical probability i.e. 1= probability(heads up) + probability(heads down). Physically, heads up and heads down cannot occur at the same time, but probabilities for the two can be added.

  Thus, it seems that one requires an expression for probability linked to the various p values of the incident, reflected and refracted photons. It is, however, not at all clear what this probability should be. At the same time, if the photon reflects, it interacts with the n1-n2 index of refraction interface and receives -2p momentum, while if it refracts, it receives p2-p1 momentum. We argue that physically such an interaction must take place within a region dx of finite size and that one cannot distinguish between p incident and p reflected  -2p or p refracted + p2-p1 in this x region. Thus, we seek a function such that F(p, x) = F(p reflected=-p,x) F(2p,x) or F(p,x) = F(p refracted, x) F(p2-p1, x). “X” must appear because one could have either the incident momentum or the final two momenta anywhere in x and one cannot distinguish between the two. F(p,x), however, defined in this way follows the product rule of an AND probability. We thus consider this as the probability required when treating 1D photon reflection-refraction from a photon-only viewpoint. In other words, we rely on momentum conservation in order to develop a probability function which is then applied to a photon only viewpoint for reflection-refraction using continuity of F(p,x) in space and continuity of its first derivative. In other words, the quantum form, exp(ipx), emerges from trying to mathematically describe momentum conservation.

 

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