Relativistic correction to Hydrogen atom - Perturbation theory

Let's think about a system that has a two-fold degeneracy for some given energy level. That is, two states $ \psi_{a} $ and $ \psi_{b} $, both of which correspond to energy $ E_{0} $. An example would be a spin-1/2 particle with a Hamiltonian that is spin-independent.

Now imagine that when we apply a perturbation, H', to the system, the degeneracy breaks into two distinct energy levels, $ E_{1} $ and $ E_{2}$.

The subtlety is that these two distinct energy levels do not necessarily correspond to the two degenerate states (it is not necessarily the case that $ H'\psi_{a} = E_{1}\psi_{a} $ and $ H'\psi_{b} = E_{2}\psi_{b} $). It is possible that the two distinct perturbed energies correspond to linear combinations of the two degenerate states, i.e. $ H'(\alpha \psi_{a} + \beta \psi_{b}) = E_{1}(\alpha \psi_{a} + \beta \psi_{b}) $, and similarly for some other linear combination (orthogonal to the first).

This subtlety is the reason that in degenerate perturbation theory, the first-order corrections to the energy require calculation of off-the-diagonal elements, i.e. things that look like $ \langle \psi_{a} | H' | \psi_{b} \rangle $. This is annoying, because in first order perturbation theory, we only needed to calculate one inner product. In degenerate perturbation theory, we have to compute a whole matrix of inner products to calculate the correction to the energy.

Since it is tedious to compute inner products, it'd be nice to know a trick to find out if the off-the-diagonal elements of the perturbing Hamiltonian are 0. The trick is given and proven on pg. 259-260 of Griffiths.

In the case of your question, the original Hamiltonian is spherically symmetric (Coulomb potential has no angular dependence). Also, the the perturbing Hamiltonian is spherically symmetric. If you look at the form of the angular momentum operator, you will note that it only involves things with $ \theta $ and $ \phi$ (derivatives and cosines and stuff). The nice thing about that, is that if I have a Hamiltonian that is purely radial (depends only on r), than it definitely commutes with L.

All Griffiths is doing is showing that = the conditions of the theorem on pg. 259 are satisfied, which shows that degenerate perturbation theory collapses to non-degenerate perturbation theory.

I'm not familiar with the idea of states being "connected" by a Hamiltonian. I would speculate that two states are connected by a Hamiltonian if the expected value of one state is nonzero given that it is initially in the other state.

Hope this helps!