Numerical Values for Hydrogen Fine Structure

Abstract

Bethe's "average excitation potentials" for states $1s$ through $4p$ of hydrogen are calculated numerically. These lead to values of the $n_0{}_{}{}^2{}_{}{}^{}S_{\frac{1}{2}}^{}{}_{}{}^{}-n_0{}_{}{}^2{}_{}{}^{}P_{\frac{1}{2}}^{}{}_{}{}^{}$ level shifts in hydrogen for $n_0=3\mathrm{} \mathrm{and} 4$ of ${\mathcal{S}}_{\mathrm{H}}\mathrm{}\left(3\right)=314.690\mathrm{}\pm 0.047$ Mc/sec and ${\mathcal{S}}_{\mathrm{H}}\mathrm{}\left(4\right)=132.998\mathrm{}\pm 0.020$ Mc/sec, which are 1.44 Mc/sec and 0.85 Mc/sec larger than previous estimates. For $n_0=2\mathrm{}$, a value of ${\mathcal{S}}_{\mathrm{H}}\mathrm{}\left(2\right)=1057.21\mathrm{}\pm 0.16$ Mc/sec is obtained, which is in close agreement with previous calculation. In conjunction with this work, the oscillator strengths $f(n_0{l}_0, nl)$ are obtained for $n_0=1, 2, 3, \mathrm{and} 4$; $n<\mathrm{}50\mathrm{}$; and all values of $l_0$ and $l$. A generalization of the Wigner-Kirkwood sum rule is derived.