Linear confinement correction to Coulomb bound states ($\Delta E_{nl}=b⟨r{⟩}_{nl}$)

For reduced mass $\mu$ and Coulomb strength $a$, put $a_q={\hslash }^2/(\mu a)$. The perturbation $br$ shifts the $1s,2s,2p$ energies by $3b{a}_q/2,6b{a}_q,5b{a}_q$, respectively. These values follow by integrating $r^3|R_{nl}{|}^2$. Because the perturbation is central, angular orthogonality makes its matrix diagonal inside each Coulomb energy eigenspace. It splits the accidental degeneracy between different orbital angular momenta while preserving magnetic degeneracy.