Let's say you have some quantum particle whose eigenvalues for some observable Q are either degenerate or non-degenerate. If you measure the observable and find it to be in a non-degenerate state, then you know that the wave function has collapsed onto this state. Now if you measure the observable and find it to be in a non-degenerate state, does this mean that the particle's wave function has collapsed to a superposition of all states sharing this eigenvalue? My answer before writing this was "I have no idea," but my answer afterwords was "Of course it does." I'd still like to see if my latter answer was wrong, because that would mean something interesting is going on.