I am aware of the debate on whether Schrödinger equation was derived or motivated. However, I have not seen this one that I describe below. Wonder if it could be relevant. If not historically but for educational purposes when introducing the equation.
Suppose that we have the time dependent Schrödinger equation for a free particle, $V=0$.
$$-\frac {\hbar i}{2m} \nabla^2 \Psi_\beta = \frac {\partial \Psi_{\beta}}{\partial t} $$
As the particle moves its heat is diffused throughout space. Now consider that we consider Heat equation or in general Diffusion equation:
$$\alpha\nabla^2 u= \frac {\partial u}{\partial t} $$
Where $u$ is temperature.
Also we have particle diffusion equation due to Fick's second law.
$$D \frac {\partial^2 \phi}{\partial x^2}= \frac {\partial \phi}{\partial t} $$
Where $\phi$ is concentration.
Furthermore, probability density function obeys Diffusion equation. So as the free particle moves, the heat, the temperature, or the density is diffused.
Now we can motivate Schrödinger equation in an intuitive way. Mathematically it is describing the same diffusion. Am I right? Have you seen more like this motivation elsewhere?
Answer
I don't know whether Schrödinger proved or guessed the equation with his name, but this equation can be derived similarly with the diffusion equation - see Gordon Baym, "Quantum Mechanics".
However, differently from the diffusion equation, the diffusion coefficient in the Schrodinger equation is imaginary. That tells us that we have to separate the Schrödinger equation into two, one equating the real parts of the two sides, and one equating the imaginary parts. The meaning of this imaginary diffusion coefficient is therefore that the wave-function is complex, or, in other words, it has an absolute value and a phase, like the electromagnetic wave.
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