I read through the derivation of bound and scattering states for a finite square well. The logic made sense to me, but I am not entirely sure how to accommodate an arbitrary initial wave function (with mean E < 0). Afterall, there are only a finite number of bound states. My guess was that the initial state might be constructed as a (unique?) mixture of both bound and scattering states. But how could this build a state with energy less than that of the lowest bound state? Could someone clarify? thx.