This is for an introductory QM course. Say, I have pure states $\vert\alpha\rangle$ and $\vert\beta\rangle$. I would like to find a unitary operator that takes me from one to the other. What is the obvious way to do this? I also realize that the constraints are not fully specified but that's fine.
If I try $U = \vert\beta\rangle\langle\alpha\vert$, this seems to do the job but is not unitary. Other approaches seem quite messy e.g. the answer at Unitary transformation from pure unentangled state to pure entangled state (nothing to do with entanglement).
What is an efficient way to find the requisite $U$ given the states in question?