When the potential function is an even function, the solution of the Schrodinger equation can be chosen to be an odd function or an even function. I know how to prove this for certain examples, e.g. this & this Phys.SE posts.
Additionally I observed that if the ground state function is even, then the 1st excited state is odd, the 2nd excited state is even, etc. in every example problem that I solved. I would like to know when this holds, and why this holds.
