Continuity Equation Versus Free Particle Flow/Flux Quantum Equations

  The continuity equation:  d/dt partial d(x,t) + grad dot d(x,t)u(x,t) = 0 ((1)) where d(x,t) is density and u(x,t) a velocity field appears in the literature as a fundamental equation of “continuity”. For example, it follows from Boltzmann’s transport equation (and conservation of mass) and leads to a wave equation for sound when combined with a pressure equation (1). A continuity type equation also follows from Maxwell’s em equations and includes the Poynting momentum b El x B where b is a constant, El electric field and B magnetic field. This equation describes light. Finally the continuity equation is used in a number of derivations of the Schrodinger equation (e.g. (2)). 

    In previous notes (3) we have argued that two flux/flow equations for free particles:  d/dx partial A = p ((2a)) and d/dt partial A = -E ((2b)), with A=-Et+px being a solution, hold for both quantum particles with rest mass and without (e.g. photons/phonons). These equations are also involved with flux which begs the question: What is the relationship between the two?

   We investigate this in this note.