This Quantum World/Implications and applications/Time independent Schrödinger equation

Time-independent Schrödinger equation

If the potential V does not depend on time, then the Schrödinger equation has solutions that are products of a time-independent function $\psi(\mathbf{r})$ and a time-dependent phase factor $e^{-(i/\hbar)\,E\,t}$:

$\psi(t,\mathbf{r})=\psi(\mathbf{r})\,e^{-(i/\hbar)\,E\,t}.$

Because the probability density $|\psi(t,\mathbf{r})|^2$ is independent of time, these solutions are called stationary.

Plug $\psi(\mathbf{r})\,e^{-(i/\hbar)\,E\,t}$ into

$i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial}{\partial\mathbf{r}}\cdot\frac{\partial}{\partial\mathbf{r}}\psi + V\psi$

to find that $\psi(\mathbf{r})$ satisfies the time-independent Schrödinger equation

$E\psi(\mathbf{r})=-{\hbar^2\over2m}\left(\frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}\right)\psi(\mathbf{r})+V(\mathbf{r})\,\psi(\mathbf{r}).$