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 $\psi (𝐫)$ and a time-dependent phase factor $e^{-(i/\hslash )Et}$:

$\psi (t,𝐫)=\psi (𝐫)e^{-(i/\hslash )Et}.$

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

Plug $\psi (𝐫)e^{-(i/\hslash )Et}$ into

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

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

$E\psi (𝐫)=-\frac{{\hslash }^2}{2m}(\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2})\psi (𝐫)+V(𝐫)\psi (𝐫).$