# Some questions about perturbation theory in QM

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.

Using orthonormal states in the degenerate subspace (and in general) simplifies enormously all computations. When we work with a self-adjoint operator (like $H$ and all their order by order pieces, $H_0, H_1, ...$), we make use of the spectral theorem to claim that there is a basis of orthonormal vectors for our Hilbert space consistent of eigenvectors of such operator.