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On the use of the axially symmetric paraboloidal coordinate system in deriving some properties of Stark states of hydrogenic atoms and ions

While the parabolic coordinate system has been widely applied in providing the quantum mechanical basis for the description of the Stark states, following the pioneering studies of Schrödinger and Epstein of the Stark effect, use of the alternative, closely related axially symmetric paraboloidal coordinate system in atomic physics appears to have been practically confined to the so-called ‘Old Quantum Theory’ of Sommerfeld. The aim of the present study is to show that the axially symmetric paraboloidal coordinate system, which has found practical application in electromagnetic theory, can also be used quite readily for deriving the properties of the Stark states. In this paper, we consider the application of this coordinate system both from the standpoint of wave mechanics and the operator calculus of the O(4) symmetry group applied to the Coulomb problem, obtaining the analogue of the Kramers–Pasternack recursion relations, which may readily be used in the determination of coordinate matrix elements, besides examining some properties of the Runge–Lenz–Pauli vector operator, in particular in relation to the Pauli identity. The generalised momenta corresponding to the curvilinear coordinates are defined, and the canonically conjugate (Hermitian) momentum operators derived. The discussion is illustrated in conclusion by a straight-forward numerical example, which relates the present treatment to corresponding results obtained in spherical polar coordinates, for both the radial matrix elements and the radial Heisenberg uncertainty product. The use here of ladder techniques in both the coordinate matrix elements and in the projection of the state functions on different coordinate systems would again provide a ready means of efficient calculation for highly excited states in dilute astrophysical plasmas, complementing the discussion in earlier papers (Hey 2007 J. Phys. B: At. Mol. Opt. Phys. 40 4077–96; Hey 2015 J. Phys. B: At. Mol. Opt. Phys. 48 185701), when one wishes to avoid numerical problems associated with very high principal quantum numbers in the analysis of radio recombination spectra from H II regions (Hey 2006 J. Phys. B: At. Mol. Opt. Phys. 39 2641–64).