Probabilistic exp(ipx)/Momentum and Deterministic t/Energy in Reflection/Refraction

  In Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world.

   Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a  to reflect/refract:  1= a + 1-a.

   Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of  2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance.

    Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected  or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength),  one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.