Continuity Equation as One Part of Work-Impulse Dichotomy in Quantum Mechanics

Classical mechanics contains a well-known dichotomy, namely that of work: Integral Fdx = change in KE and impulse, Integral Fdt = change in p (momentum). The former view is concerned with time and velocity, while the latter is not as p=m1v1=m2v2, i.e. the same impulse may occur for two different velocities. When solving classical mechanical problems, however, the time view (conservation of energy equation, Newton’s second law) is utilized and the impulse picture, usually ignored.

   We suggest that in quantum mechanics, one must retain both dichotomies as they are linked to each other as they are in classical physics. In fact, we argue that in quantum mechanics interactions are based on the impulse picture, which in some problems, such as one dimensional photon reflection-refraction, make it the only scenario which solves the problem. 

(Actually the impulse scenario makes use of flux values and so is intertwined with the time picture.)

    In the literature (1) etc, the continuity equation:  d/dt partial density = d/dx { velocity(x) density(x,t) } seems to be a common starting point for deriving the Schrodinger equation. We argue that this approach only considers the time-work-kinetic energy scenario a priori and ignores, a priori, the impulse approach to interactions. The impulse picture is a physical description of interactions and so both scenarios should be introduced together a priori. What is usually done is to consider only the time picture and then  hypothesize on mathematical grounds that:  density(x,t) = W*(x,t)W(x,t) instead of considering the wavefunction form as the representation of the impulse picture which governs interactions. In fact, the time-independent Schrodinger equation is a combination of both scenarios. The work-kinetic energy picture must appear because V(x) is used, but actual interactions occur via impulses and their unusual associated exp(ipx) probability.