Editing Heat equation (section)

=== Schrödinger equation for a free particle ===
{{main|Schrödinger equation}}

With a simple division, the [[Schrödinger equation]] for a single particle of [[mass]] ''m'' in the absence of any applied force field can be rewritten in the following way:
: <math>\psi_t = \frac{i \hbar}{2m} \Delta \psi</math>,
where ''i'' is the [[imaginary unit]], ''ħ'' is the [[reduced Planck constant]], and ''ψ'' is the [[wave function]] of the particle.

This equation is formally similar to the particle diffusion equation, which one obtains through the following transformation:
: <math>\begin{align}
c(\mathbf R,t) &\to \psi(\mathbf R,t) \\
D &\to \frac{i \hbar}{2m}
\end{align}</math>

Applying this transformation to the expressions of the Green functions determined in the case of particle diffusion yields the Green functions of the [[Schrödinger equation]], which in turn can be used to obtain the [[wave function]] at any time through an integral on the [[wave function]] at ''t'' = 0:
: <math>\psi(\mathbf R, t) = \int \psi\left(\mathbf R^0,t=0\right) G\left(\mathbf R - \mathbf R^0,t\right) dR_x^0 \, dR_y^0 \, dR_z^0,</math>
with
: <math>G(\mathbf R,t) = \left( \frac{m}{2 \pi i \hbar t} \right)^{3/2} e^{-\frac {\mathbf R^2 m}{2 i \hbar t}}.</math>

Remark: this analogy between quantum mechanics and diffusion is a purely formal one. Physically, the evolution of the [[wave function]] satisfying [[Schrödinger's equation]] might have an origin other than diffusion{{citation needed|date=January 2023}}.