According to the lecture I'm hearing the SE can be simplified for the cases of a timeinvariant potential and a constant potential:(adsbygoogle = window.adsbygoogle || []).push({});

Time invariant

[itex]\frac{\partial^2\psi}{\partial x^2} = 2\frac m{\hbar^2}(V(x)-E)\psi[/itex].

Then, the lecture states that for the case of the potential being not just time but also spatially invariant it can be simplified to

[itex]\frac{\partial^2\psi}{\partial x^2} = -k^2\psi[/itex]

My question is, why this is said to be possible, only if the potential is constant. Given the term [itex]V(x) - E[/itex] we can conclude that [itex]V(x) - E = E_{kin}[/itex] and hence [itex]\frac12\hbar^2\frac{k^2}m[/itex]

Thanks

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# Special cases of the Schroedinger Eq?

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