The N eigenvalues obtained by solving this equation give the shifts in the degenerate energy level due to the applied perturbation, while the eigenvectors give the perturbed states in the unperturbed degenerate basis | m ⟩ ⟩ . To choose the good eigenstates from the beginning, it is useful to find an operator V ^ ^ which commutes with the original Hamiltonian H 0 ^ ^ and has simultaneous eigenstates with it.