Schroedingers equation with gauge coupling derived from a continuity equation

We consider a statistical ensemble of particles of mass m, which can be described by a probability density ρand a probability current \vec{j} of the form ρ\nabla S/m. The continuity equation for ρand \vec{j} implies a first differential equation for the basic variables ρand S. We further assume that this system may be described by a linear differential equation for a complex state variable χ. Using this assumptions and the simplest possible Ansatz χ(ρ,S) Schroedingers equation for a particle of mass m in an external potential V(q,t) is deduced. All calculations are performed for a single spatial dimension (variable q) Using a second Ansatz χ(ρ,S,q,t) which allows for an explict q,t-dependence of χ, one obtains a generalized Schroedinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schroeodingers equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for an non-unique external q,t-dependence of χ, one obtains Schroedingers equation with electromagnetic potentials \vec{A}, ϕin the familiar gauge coupling form. A possible source of the non-uniqueness is pointed out.

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