Hong-Ou-Mandel Effect and One Dimensional Reflection-Refraction

We have suggested in previous notes that exp(ipy), the free particle wavefunction, is really a complex probability (same modulus for any free particle p momentum) which allows for equal weights for any two AND (product) momenta in a 2-body scattering scenario if momentum is conserved, i.e. exp(ip1y)exp(ip2y) = exp(ip3y)exp(ip4y) if p1+p2 = p3+p4 (momentum only in the y direction). This product AND scenario applies to two particles, but a single particle may be linked with an OR scenario which adds exp(ipy)s for the same particle as in a 2-slit interference calculation if the slits are about hbar/p apart. These ideas may be applied to one dimensional reflection-refraction (which is discussed in (2)). If one thinks of exp(ipy) as representing a probability, then an incident and reflected photon (same photon) interferes with itself as interference is simply OR probability (add) with time removed. The reason time may be removed is that this calculation may occur at the interaction point in which case the incident, reflected and refracted photon all exist at roughly the same time and Heisenberg uncertainty ensures that one cannot pinpoint an incident or reflected photon at an exact x and t. One then uses continuity of probability exp(ipy) and the derivative d/dy exp(ipy) to obtain the results of (1). We note that if one considers the case of a very tiny transmission, then the photon is essentially reflected, but there are still two cases n1