- #1

Happiness

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- TL;DR Summary
- <1/r^3> uses the standard wavefunctions ψ_nlm of hydrogen, which are not good states to use in perturbation theory because the Hamiltonian (under spin-orbit interaction) no longer commutes with L. So shouldn't we solve for the simultaneous eigenstates of L^2, S^2, J^2 and J_z first? And then use those to find <1/r^3>?

Below is the derivation of E

For elaboration, the phrase "good state" relates to the following theorem:

Ordinary first-order perturbation theory means using [6.9] below, with ##\psi^{0}_{n}## replaced with a good state.

^{1}_{so}, the first-order correction to the Hamiltonian due to spin-orbit coupling of the election in hydrogen atom. My question is whether it's valid to use [6.64] (see below). ##<\frac{1}{r^3}>## I believe is ##<\psi_{nlm}|\frac{1}{r^3}|\psi_{nlm}>##, but ##\psi_{nlm}## is NOT a good state to use in perturbation theory, because ##\psi_{nlm}## is an eigenstate of ##L_{z}## but H'_{so}does not commute with ##L## (as mentioned in the paragraph above [6.62]-[6.63]).For elaboration, the phrase "good state" relates to the following theorem:

Ordinary first-order perturbation theory means using [6.9] below, with ##\psi^{0}_{n}## replaced with a good state.