I read the following text form physics forum. How do you formally derive Schrodinger equation this way?
(The Feynman-Kac formula)
A Wiener process represents Brownian motion. Brownian motion has two terms: viscosity and N-dimensional Gaussian noise. Viscosity we set to zero. The variance of the N-Gaussian we calculate from Planck's constant. Often something called drift is added in. This is a constant momentum and may be set to whatever you like.
A Wiener process with no drift has a spectral representation as a sine series whose coefficients are independent N(0,1) random variables. So that's where you get the "uncertainty." Now combine this random spectrum with the drift, the mean momentum term we added in. Take the inverse Fourier transform using the drift time as the variable of integration and you get the Schroedinger wave function. Cool!