If I want to calculate perturbed energy states in 2-fold degenerate case in Quantum Mechanics.
Assume the Hamiltonian is $ H=H_0+H^\prime$
Then we can calculate matrix elements of W : $ \langle a|H^\prime|b\rangle$, where a and b are eigenvectors which span the degenerate subspace.
And by solving the characteristic Eq of W, we can get pertured energy states.
My questions are:
(1) Should we always take $i$ and $j$ to be orthogonal? In Griffiths' book the derivation of time-independent theory assumed orthonormality.
(2) Can any orthonormal states in subspace be "good states" ? Because in his book $|\psi_a\rangle$ and $|\psi_b\rangle$ are arbitrary and indeed good states.