Quantum Mechanics and Brownian/Stochastic Motion Part II

As mentioned in previous notes, there exist derivations of the Schrodinger equation in the literature based on stochastic equations and spatial density. In this note, we focus on a particular derivation (1) which makes use of osmotic velocity using an operator which describes a Brownian type motion. The Schrodinger equation follows in (1) from a Newton type equation with d/dt being replaced by an operator which tracks stochastic motion. In particular, the spatial density is set equal to W*(x,t)W(x,t) at the last moment, where W(x,t) is a complex function.

   We compare the approach of (1) to one in which statistical ideas are applied from the beginning to obtain a conditional probability P(p/x)=a(p)exp(ipx)/W. This together with KEave(x) + V(x) = E where KEave(x)=Sum over p  pp/2m P(p/x) yields the time-independent Schrodinger equation.

    We find that the two approaches are consistent, but that in the approach of (1), one does not use the wavefunction until the final steps.