It is straightforward (and hence left for the reader :-) to prove that all off-diagonal elements of the set (49) are equal to 0 . Thus we may use Eq. (27) for each set of the quantum numbers \(\{n, l,...It is straightforward (and hence left for the reader :-) to prove that all off-diagonal elements of the set (49) are equal to 0 . Thus we may use Eq. (27) for each set of the quantum numbers {n,l,m} : \[\begin{aligned} E_{n, l, m}^{(1)} & \equiv E_{n, l, m}-E_{n}^{(0)}=\left\langle n l m\left|\hat{H}^{(1)}\right| n l m\right\rangle=-\frac{1}{2 m c^{2}}\left\langle\left(\hat{H}^{(0)}-\hat{U}(r)\right)^{2}\right\rangle_{n, l, m} \\ &=-\frac{1}{2 m c^{2}}\left(E_{n}^{2}-2 E_{n}\langle\hat{…
