• The perturbed eigenstates of $$\hat{H}$$ are linear combinations of degenerate eigenstates of $$\hat{H}_0$$. This means that they too are eigenstates of $$\hat{H}_0$$ from a different eigenbasis.
• If $$\hat{H}_0$$ is compatible with $$\hat{V}$$, i.e. $$[\hat{H}_0, \hat{V} ] = 0$$, then there is no mixing with non-degenerate states and the analysis above is exact.
• Notice how the mathematics mimics the quantum mechanics. Without the perturbation the eigenbasis of $$\hat{H}_0$$ is not unique. When we try to determine its energy shift we find a matrix equation which can only be solved for specific values of $$\Delta E_j$$. These $$\Delta E_j$$ in turn correspond to specific choices for the coefficients $$c_{ji}$$, i.e. particular linear combinations of the unperturbed states. Thus to solve the equations we are forced to collapse the wavefunction onto an eigenstate of $$\hat{V}$$. $$V_{ki}$$ is a Hermitian matrix, and consequently has real eigenvalues.