When riding home today from school on my bike, I was thinking about some QM. The general solution to the Schrödinger equation for t=0 is given by:
where [itex]\psi_n(x)[/itex] are the eigenfunctions of the Hamiltonian. We know that the probability of [itex]\Psi(x)[/itex] collapsing to one of the eigenstates is given by the absolute square in front of that particular eigenstate.
Now my question comes: Lets say we have an arbitrary operator Q: [itex]Q\psi_n = q_n\psi_n[/itex], where qn are the eigenvalues. Now what do I do if I want to find the probability of [itex]\Psi(x)[/itex] collapsing to one of the eigenstates of the operator Q upon measurement of the observable Q?
My own attempt is that we write [itex]\Psi(x)[/itex] as a new linear combination of the eigenstates of Q. But to me this seems very difficult.
I hope you can shed some light on this. Thanks in advance.