How do we know/prove that a perturbation splits degenerate states into orthogonal linear combinations? On page 257 of Griffiths' Quantum book, he says "Typically, the perturbation will break the degeneracy: As we increase $\lambda$, the common unperturbed energy $E^0$ splits into two. Going the other direction, when we turn off the perturbation, the "upper" state reduces down to one linear combination of $\psi^0_a$ and $\psi^0_b$ [the degenerate states], and the "lower" state reduces to some orthogonal linear combination, but we don't know a priori what these "good" linear combinations will be."
Griffiths provides no justification for the proposition that the perturbation splits degenerate states into degenerate linear combinations, he just states it which isn't very satisfying. It feels right, but I'm not sure how to prove that this must be true. In fact, I now realize that I don't think any justification was given for the stationary states of an unperturbed Hamiltonian being an orthogonal set.
Could you please show me why this must be the case? Thank you!
 A: In general a perturbation need not lift the degeneracy of the original Hamiltonian, Griffiths simply focuses on the case where it does because the case where it does not is essentially the same as non-degenerate perturbation theory and so relatively uninteresting.
When the perturbation does lift the degeneracy the eigenstates of the perturbed Hamiltonian are, by definition, eigenstates of a Hermitian operator with different eigenvalues and so must be orthogonal.
A: Degenerate states come from a symmetry in the Hamiltonian. For instance, some degree of freedom needed to describe a state of the system might not be present in the Hamiltonian at all. This is the case for the Hydrogen atom without corrections, the spin operator does not appear in the Hamiltonian at all.
If the perturbation you are considering also shares the same symmetry then it will not break the degeneracy of the states. However, again using the example of the hydrogen atom, the degeneracy for spin states is broken when you consider a perturbation that couples the spin to a magnetic field. 
So in general, you need to see whether the symmetry of the Hamiltonian which causes the degeneracy is present in the perturbation.
