As discussed here, the quantum state $S$ that corresponds to the solution $\psi(t,x)$ of a time-independent Schrödinger equation remains constant wrt. time. I assume that the the solution of a time-dependent Schrödinger equation may then produce a state $S_t$ that does depend on time.
What is the physical significance of $S$ versus $S_t$? More specifically, does (in some sense) $S_t \to S$ as $t \to \infty$? Heuristically, this would seem logical, since states like $S$ are occasionally described to be "standing waves".
A motivating example: Consider a particle in a box, whose time independent states are well known. Measuring the location of the particle would collapse the wave function which produces a state that is not an eigenstate of the Hamiltionian. The particle is then again delocalised, hence, the state clearly also evolves wrt. time. Question 1: Is this evolution $S_t$ a solution to a time-dependent Schrödinger equation? Question 2: Does $S_t$ approach an eigenstate of the Hamiltionian as $t \to \infty$?