GRE Quetion (QM, Electric Transition)

[HungryChemist]

Homework Statement

X and Y are two stationary states of a particle in a spherically symmetric potential. In which of the following situations will the wave functions of the two states be orthogonal?
I X and Y correspond to different energies.
II X and Y correspond to different total orbital angular momenta L
III X and Y correspond to the same L but different Lz.

Homework Equations

Selection Rule for Electric Transition?

The Attempt at a Solution

This is GRE type question. I tried attacking this problem by invoking the selection rule for electric transition. I am going to say two state being orthogonal means the situation never occur. So, out of three situations stated above, which one of them has probability of zero occurring? First choice seems to occur frequently, so can't be orthogonal. Second is also not orthogonal. But the last situation, I am not sure. Can anybody help? In fact, if anybody has good web sources that treats electron transition rule in hydrogen atom (not hyperphysics) please, share the link with me. Thanks.

 

[Dick]

You are thinking like a chemist, bless your heart. But all of these conditions are eigenstates corresponding to different eigenvalues of a hermitian operator. They are all orthogonal. And transitions between any of these states can occur. If they are perturbed. If not, then not.