Quantum Mechanics and Double Conservation

We suggest that quantum mechanics is concerned with preserving two conservations, one related to a kind of probability and the other to a mechanical balance.These two conservations exist at the level of a single particle and are interlinked. Specifically, the derivative d/dx, which measures the effect of a change in a kind of probability, also measures the associated resistance or impulse for a free particle. For a free particle, the “kind of probability” is exp(ipx) and d/dx exp(ipx) = ip exp(ipx) showing that there is an impulse related to p linked with the probability. In other words, a free particle moves through space in a probabilistic manner related to the resistance associated with the impulse p. These two conservations, a kind of probability and a kind of resistance, lead to two independent equations when one considers continuity at points which change momentum.

    We consider again the problem of a photon moving in the x direction which reflects or refracts at an interface along the y axis between n1-n2 (indices  of refraction). Classically, probability is conserved i.e.  1= P(reflect) + P(refract) ((1)). The steady state problem, however, has the appearance of an equilibrium and should have a second conservation, namely one in space, i.e. a balance of pressure. This balance, linear in momentum, however, is the identical equation as ((1)). Thus, there are still two unknowns P(reflect), P(refract) and one equation. 

  In the quantum case, double conservation may classically be manifested by the same equation. This seems to suggest, we argue, that ((1)) is a consequence of an equation based on p (which is linked to momentum). We try to find this association in two ways. First, we argue that the interface may exist at xo and that there are various momenta in the problem, We argue these represent pertinent information which does not appear in ((1)). We suggest that ((1)) seems to be a kind of magnitude equation with p and xo information being part of the direction of vectors i.e. p and xo may exist in a phase exp(i f(p,x)). Together with the conservation of pressure condition, we argue that exp(i f(p,x)) = exp(ipx). This argument, however, implicity involves consider the phase function exp(if(p,x)) as a kind of probability.

   Alternatively, we consider finding this probability follows from maximizing Shannon’s entropy subject to the periodic constraint i p x P(x), but we investigate why Shannon’s entropy form should be used, i.e. what is its physical significance?