We assume that the most important similarity is that both describe how the energy density function U(x,t) moves in 4D x-t space.
On the other hand, the most important difference is that in the heat diffusion equation the energy density function can be constrained inside a box of Dirichlet boundary conditions, whereas in the Schrödinger equation, it cannot (the wave function 𝛙 extends by definition to infinity).
Of course, you can add many similarities and differences, but how?
The equations are simply relations between the flux and "delsquared" driver equations. The forms of solution are similar but the boundary conditions are different. Another issue with the Fourier equation is whether the flux can reach the speed of light, (or even exceed it) under impulse conditions. The general view on this seems to be that heat conduction has the limits of the Maxwell equation for electrical fields: approximately the speed of sound in the material.
Here is a more complete and coherent answer published on Researchgate and the IJISRT journal:
A statistical numerical solution for the time-independent Schrödinger equation-II, November 2023.
Your comments on the article are welcome.
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