Quantum Mechanics and Brownian/Stochastic Motion

 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.