The transition from Classical to Quantum Mechanical Physics

The top diagram shows a collection of particles on the left side of a divider.  If you imagine removing the divider the motion in aggregate is going to be described by a decaying exponential with a real term in the exponent. 

The lower diagram shows the same setup with only one particle.  Since we intend on looking at it in particular it is obvious that a decaying exponential will not be sufficient.  Once the particle manages to travel from the left half of the divider to the right half of the divider it has the same level of chance to return to the left side.  Thus it requires some sort of imaginary term in the exponential if we assume that diffusion still reigns as the mechanism of transport

In the Feynman lecture on physics there is a derivation of the Schrodinger wave equation that assumes difusion of the type in the second image.

Feyman Lectures on Physics Online