Quantum Time Removal Within a Wavelength and Reflection/Refraction

The problem of an incident beam of light traveling to the right in the x direction and reflecting/ refracting at an interface at x=0 may be solved using Maxwell’s electromagnetic equation for the electric field of a photon. In particular, mathematical continuity of the electric field and its first derivative, where the field is of the form Real( exp(ipx - iEt) ) with E energy being  constant, but p allowed to change, may be used to establish relationships between weights of the incident, reflected and refracted electric fields. The modulus of each of these waves is then linked to intensity. 

   The above continuity equation, however, is somewhat unusual if one thinks of a single quantum traveling in time. For instance, before it strikes it is an incident photon, but after (in time) it is either a reflected or refracted one. Thus probability naturally appears in the problem and if one performs the usual grouping by time, then the incident photon is one side and the reflected or refracted on the other. This is the opposite grouping used in the continuity approach so we ask: How may this be understood? 

      On average a photon moves with a certain speed, but it also has a wavelength. We argued in (2) that for a particle with rest mass, there is no time within the wavelength region, but rather a probability to be at an x value is given by exp(ipx). In fact, the two dimensional nature of exp(ipx) accounts for the removal of time as one needs two “variables” or dimensions to describe the direction of motion. On average, however, the particle still moves as x=velocity time. If there is no time in the probabilistic approach within a wavelength, then even treating the particle as a point means that there is no clear separation in time of the incident, reflected and refracted rays. If one uses space as a marker (as in the time-independent Schrodinger approach to a bound quantum particle) then this spatial approach within a wavelength governs the solution to the problem.

    In such a case, one may write Aexp(ipx) + B exp(-ipx) = Cexp(i p2 x) and a similar conservation of momentum equation A p exp(ipx) -p B exp(-ipx) = Cp2 exp(ip2 x) rather than argue that the electric field and its derivative should be continuous at x=0. We use addition because there exists an OR situation, but we do not employ the usual grouping by time, but rather in space with this probability.On average, however, this approach must be consistent with AA/c1 = BB/c1 + CC/c2 where A*A etc are intensity values much like W*(x)W(x)=Probability(x) for a bound particle where W(x) is the wavefunction.

  Mathematically the above approach is the same as the continuity of the electric field and its first derivative, but we suggest that this view considers the role of time in the problem. The continuity approach allows for the incident, reflected and refracted photon to exist at the same time which only makes sense if time does not really exist within a wavelength, even though there is a mapping to x=vt. Thus both the loss of time and the idea of probability are needed and they complement each other.