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

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[edit] 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}).

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