Schrodinger Equation from Fokker-Planck Equation and Contradiction Due To Interference?

In (1) it is shown how the Schrodinger equation can be “forced” in a sense to follow from the Fokker-Planck classical stochastic equation which contains a classical probability density. Other approaches in the literature which are similar use continuity equations and also a classical density. The key to all of these approaches is to assume, in an ad hoc manner it seems, that classical probability is equivalent to W*(x)W(x) where W(x) is a complex function called the wavefunction. In principle, one may write a density as W*(x)W(x). This is mathematically correct even if, for the sake of argument, it is ad hoc. The problem is that for a constant density (i.e. constant momentum and speed), one does not know if W(x)=exp(ipx) or W(x)=exp(-ipx).

      We consider here the arguments shown in (1). If a free particle has a density(x)=1/L, it may be associated with W(x) = exp(ipx)/sqrt(L).  A free particle density, however, is associated with constant p (momentum) and assumes one knows x(t). If one considers the free particle to be trapped in a one-dimensional box with infinite potential walls, it travels to the right with p and then to the left with -p. The density(x) classically is 1/L, where L is the length of the box. The density is constant in time, but the direction of motion is a function of time. Thus, information is lost considering an equation such as the Fokker-Planck equation which uses a density function which does not track the direction of motion in time. In particular, writing density(x) = W*(x)W(x), which is done in (1) and applying this density to a bound particle in a box with infinite walls leads to an equation linear in W(x).

  Thus exp(ipx) and exp(-ipx) are both possible solutions if V(x)=0. Given that the conservation type equation obtained from the Fokker-Planck (or continuity) equation is linear in W, exp(ipx)+exp(-ipx) is a solution. Thus there is interference which leads to a density which does not match a constant classical density which should apply to a particle moving to the right and then to the left in a bound state because one assumed a classical picture a priori. Simply suggesting that density classical = W*(x)W(x) mathematically does not mean that the a priori classical picture may be dropped. As a result, we argue that there exists a contradiction in approaches which write classical density as W*(x)W(x) and then obtain the Schrodinger equation. 

   As a remedy, we suggest that W(x) does not follow from a theory of classical density=W*(x)W(x), but vice versa. One needs equation which deal with W(x) a priori, and once it is found, one may create a classical density through W*(x)W(x).