Playing around with the Schrödinger equation, I separated the time partial derivative this way:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}-\frac{i}{\hbar}V\Psi$$
Looking at it, I realized that, for a free particle, we have:
$$\frac{\partial \Psi}{\partial t}=\frac{i\hbar}{2m}\frac{\partial^2\Psi}{\partial x^2}$$
If we call $\frac{i\hbar}{2m}=\alpha$, then:
$$\Psi_t=\alpha\Psi_{xx}$$
Which is exactly in the form of the heat equation: $u_t=\alpha u_{xx}$. Furthermore, the heat equation often contains added terms that indicate a heat loss (or gain), for example: $u_t=\alpha u_{xx}-\beta u$
Comparing to the full Schrödinger equation at the start of this post, it's clear that if $\beta=iV/\hbar$ then the Schrödinger equation is: $\Psi_t=\alpha \Psi_{xx}-\beta \Psi$. Exactly in the form of the heat equation above.
So what's going on here? These equations are too similar for it to be a coincindence: are there fundamental reasons as to why this is the case?
