Classical mechanics, the diffusion (heat) equation, and the Schrödinger equation

We consider the limiting case ??0 of the Cauchy problem (Equation Presented) u_?(x,0) = exp[-S₀(x)/?] T₀(x); S₀, T₀ independent of ?, for both real and pure imaginary ?. We prove two new theorems relating the limiting solution of the above Cauchy problem to the corresponding equations of classical mechanics (Equation Presented) These relationships include the physical result quantum mechanics ? classical mechanics as ??0. Copyright © 1977 American Institute of Physics.