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So take

[tex]\psi(\mathbf{r},t) = \psi_1(x) \psi_2(y) \psi_3(z) \phi(t)[/tex]

and substitute it into the time-dependent Schrödinger equation. For the stationary states set U=0 and obtain

[tex] \frac{-\hbar^2}{2m} \nabla^2 \psi(\mathbf{r},t) = i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}[/tex]

Then divide by the wavefunction, and I get

[tex]i\hbar \frac{\partial \phi(t)}{\partial t} = \frac{-\hbar^2}{2m} \left( \frac{\partial^2 \psi_1}{\partial x^2} + \frac{\partial^2 \psi_2}{\partial y^2} + \frac{\partial^2 \psi_3}{\partial z^2} )\right [/tex]

I know that each one of the unknown functions must make a separate equation, but I don't know what to solve for without energy. For the time-independent equation they will all essentially be infinite square wells, but I don't know what to do with the time dependency.