Inference of the Doebner-Goldin Nonlinear Schrödinger Equation from Classical-Mechanics Solution. The Implied Applications
We set up the classical wave equation for the total wave of a particle traveling in a conservative potential V and a non-conservative potential of frictional force f. This separates out a component equation describing the particle kinetic motion, which for f = 0 is the usual linear Schrödinger equation as obtained in a separate work. The f-dependent probability density presents generally an observable diffusion current of a real diffusion constant D, which together with the particle process make up an adiabatic system whose probability density current is conserved. The corresponding two extra terms in the Schrödinger equation are found to be identical to the Doebner-Goldin nonlinear terms. The observable diffusion may effectively describe the heat current of a dissipative macroscopic system. But even interestingly and of wide-appeal to applications we show in terms of solutions for particle that, it may describe literally a radiation (R) depolarization! (D) field, which is always produced by the electromagnetic fields from which our particle is made, in a dielectric medium. The RD and radiation magnetic fields produce an attractive radiative Lorentz force which when regarding the vacuum as a dielectric against a true empty space, we showed earlier resembles directly Newton's universal gravity. The observable diffusion is indicated here to be intrinsically associated with the particle process, a property which was strikingly conjectured in the Doebner-Goldin original discussion.