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

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