The Schrodinger Equation and Two Stochastic Considerations

  In this note, we argue that the Schrodinger equation for one particle represents ensemble averages taken from stochastic momentum values of a single particle at different times, with time being removed. A new time, representing the evolution of the ensemble is then introduced especially for the case in which energy is being added or removed from a system (e.g. V(x,t)). The momentum stochastic features arise from the potential (or force) acting in a stochastic manner leading to a distribution of momentum values at x (at different times). If V(x,t), we argue the system adds or removes some average energy at x at t. We suggest that in reality, stochastic amounts of energy are added or removed (as long as the average is obtained). This leads to a second distribution, namely one in energy being established.. Thus, W(x,t) may be expanded in terms of exp(ipx) for a fixed t or in exp(i Ei t) for a fixed x. As a result, one may calculate average energy using either Average (p*p/2m) + V(x,t) or  [Sum over Ei   exp(i Ei t) Ei ] / W(x,t). Equating the two stochastic approaches yields the Schrodinger equation.