Information, Quantum Probability and Reflection/Refraction Part V Continuity Equation

Note: Dec. 28, 2022 The form exp(-iEt)exp(ipx) for both the photon and a particle with rest mass may be obtained from the Lorentz in variant -Et+px=A by noting that -id/dx A = p and that exp(ipx) is an eigenfunction of exp(ipx). 

  In a series of notes we have considered the problem of light moving in the positive x direction hitting a medium along the y axis at x=0 and reflecting and refracting. We assumed that light is of the form of a photon which has an energy E which is the same in the incident, reflected and refracted beams. We also considered the notion of a steady state stream. Given a single photon in the incident beam, it has a probability a to reflect and 1-a to refract. 

   As discussed in previous notes, one may consider an intensity AA, BB, CC for the incident, reflected and refracted rays (AA>0 etc) and write energy conservation as:  AA/c1 = BB/c1 + CC/c2 where c1 is the speed of light in medium 1 and c2 in medium 2. We noted that the energy conservation equation has two unknowns B,C if A is known, but may be broken into two equations if one considers time removed from the problem (i.e steady state) and one groups by space i.e.  (AA/c1 -BB/c1) = CC/c2 i.e. A+B=C and A/c1 -B/c1=C/c2. One may note that one does not have a conservation of flux/intensity equation, but that flux/intenisty which physically exists is directly linked to energy conservation. One may also note that one does not have an equation of energy flux conservation.

   In this note, we consider the dispersion relationship of light E=pc and the notion of a continuity equation which applies to the “square root” probabilities. We consider exp(-iEt+ipx) as the simplest solution, other than a constant and speculate whether exp(ipx) for a steady state picture represents a kind of square root flux. The idea of a square root flux suggests two sets of equations with square root fluxes which may be multiplied to create a flux equation. exp(ipx) has a kind of wavelike structure with two dimensions and may interfere. It also maps to a constant flux through exp(-ipx)exp(ipx). We consider these ideas and conclude that the wavefunction is like a square root flux which helps to associate a picture with the idea of it being a dynamical probability. This is helpful when considering a W(r vector) not equal to W(r) i.e. not spherically symmetric. This suggests unbalanced flux and rotation. In quantum mechanics, one does not have a container and so the fluxes create a static-appearing density in the bound case.