Abstract
A new method is proposed for computing the phase shifts and normalizations of the spherical wave functions of an unbound Dirac electron in the electric field of a point or finite-sized nucleus. The method uses the phase-amplitude formsG(r)=A cosϕ andF(r)=A sinϕ for the radial factors. The first step is to computeϕ(r 1), wherer 1 is larger than the nuclear radius but is still reasonable for machine computations in many cases of interest. This first step requires a numerical solution of radial differential equations fromr=0 tor=r 1. The second step is to compute the phase shift and a normalization correction factor directly from a new asymptotic series, whosen-th term is of the formP(ϕ)r 1 − , whereP(ϕ) is a finite Fourier series inϕ(r 1). Formulae for upper bounds on the error in the series are given and are used to estimate suitable values ofr 1 for several cases. The series is especially suitable for problems, like the decay of a μ-mesic atom, where the wave functions must be computed in any case up to distances ≳r 1. No comparison is made with other methods.
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