# Time evolution of quantum state after an observation

As described in the wikipedia article on wave function collapse, the mathematical formulation of quantum mechanics postulates that wave functions change according to two processes:

1. When not being observed, a wave function will evolve (deterministically) according to some relevant differential equation, e.g., Schrödinger's equation.
2. Upon measurement, a wave function will collapse (probabilistically) to one of its component eigenfunctions.

What I'd like to understand is how quickly after a measurement a particle's wave function will wander away, according to (1), from whatever eigenfunction it collapsed to.

As a concrete example (I hope!), consider two Stern-Gerlach experiments set up in sequence, both arranged to measure the z component of spin, in which we feed to the second experiment only the spin up half of the beam from the first experiment. When one learns about this in an introductory QM course, it is said that the observed output from the second experiment will still be all spin up. But is this still true if the experiments are spaced really far apart? How will the time evolution in (1) affect the outcome then? (Or is there something I have missed that prevents time evolution from affecting the intermediate beam?) If the observed outcome from the second experiment does depend on the distance between the experiments, then it seems this would give a way to measure (at least in principle) how rapidly a pure eigenstate can degenerate (by measuring, for example, the distance between the two experiments needed to get an output beam from the second experiment which is 75% up, 25% down).

It depends on the Hamiltonian, i.e., the interactions the beam is subject to after the first measurement. If the beam is allowed to propagate freely, then a pure state, say $|0\rangle$, will remain in that state always because the Hamiltonian only contains a momentum operator that will not affect the spin component of the state vector. If on the other hand, after the first measurement, there are other interactions like a magnetic field, then it can lead to a non-trivial Schrodinger evolution of the pure state (spin part) which will change the outcome at the second measurement.