When is Aexp(i(kx - ωt)) a solution to the Schrödinger equation?

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cytochrome
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I know that one solution to the Schrödinger equation is

Aexp(i(kx - ωt))

Is there a special relationship between k and ω for this to be a solution, or is it always a solution?
 

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That's the solution for a free particle. You can work out the dispersion relationship easily.

The energy is only kinetic, p = hk (that should be h bar), E = hω (again, h bar).