Discrete basis-set approach for calculating Bethe logarithms

We present a discrete basis-set technique for calculating Bethe logarithms for eigenstates of the hydrogen atom. Although our initial results for the 1s and 2s states are accurate to only about 2%, a high-order Neville-Richardson extrapolation yields Bethe logarithms accurate to better than 1 part in ${10}^{13}$, some three digits more precise than the best previous calculation. The dramatic improvement in accuracy illustrates the power of Neville-Richardson extrapolation, which may well be found useful for improving the accuracy of numerical calculations in other areas of atomic physics.