On the Pasternack–Sternheimer theorem for bound states in hydrogenic atoms and ions derived by operator calculus

Abstract

The Pasternack–Sternheimer theorem for bound states (Pasternack and Sternheimer 1962 J. Math. Phys. 3 1280) is obtained directly by the methods of operator calculus set out earlier (Hey 2006 J. Phys. B: At. Mol. Phys. 39 2641–64), on the basis of the factorisation technique of Infeld and Hull (Infeld and Hull 1951 Rev. Mod. Phys. 23 21–68). The present derivation, which complements the group theoretical treatments of Armstrong (Armstrong 1970 J. Phys. Colloq. 31 C4-17–23) and Cunningham (Cunningham 1972 J. Math. Phys. 13 33–9), not only elucidates the original result in terms of fundamental quantum mechanical theory, but also reveals some apparently new inter-connections between different radial matrix elements (for given n, diagonal and off-diagonal in ℓ, ℓ′) of hydrogenic atoms and ions. The key equation used to derive the theorem here is shown to follow identically in the non-relativistic limit from the treatment of the generalised Kepler problem by Crubellier and Feneuille (Crubellier and Feneuille 1971 J. Physique 32 405–11). This work is a continuation of studies employing operator methods to provide results of potential usefulness for spectroscopic studies of laboratory and astrophysical plasmas, in particular to transitions between states of high principal quantum number, as in the high-n radio recombination lines (Hey 2013 J. Phys. B: At. Mol. Opt. Phys. 46 175702; Peach 2014 Adv. Space Res. 54 1180-83).