Phase Shift of exp[i <p at x> dx] for Bound Quantum Systems Part 2

In a previous note, we suggested that the change (d/dx) in the relative conditional probability at x i.e. exp(ikx) of a photon or free quantum particle (from deBroglie’s picture) may be applied to a bound state average relative conditional probability W(x) (the wavefunction) so that W(x+dx) = W(x) exp(i dx) where = -i d ln(W) / dx. We went on to argue that - d/dx d/dx W / W = which may be linked to a conservation of energy equation i.e. the Schrodinger equation. In this note, we extend these ideas to the relativistic Klein-Gordon case and also examine further statistical properties which follow, including the result that W(x+dx)= W(x) = exp(i .5dx+ i .5dx