On the determination of radial matrix elements for high-n transitions in hydrogenic atoms and ions

Abstract

With the aid of the factorization method developed by Schrödinger, and Infeld and Hull, we use ladder operators to derive a set of recurrence relations from which electric multipole transition (radial) matrix elements of the type ⟨n′ℓ′|rk|nℓ⟩ in hydrogenic atoms and ions can be obtained with the knowledge of a few particular integrals. These ‘initial values’ for the recurrence relations are very simply derived by elementary algebra, with the aid of the properties of the associated Laguerre function. As a check of these recurrence relations, special cases of transitions with n = n′ are considered, and thereby some apparently new matrix element relations are found. This method of computation, first proposed by Infeld and Hull, is efficient, highly accurate also for very large values of the principal quantum number, and generates no numerical instability as n and n′ both grow large. As illustrations of the method, electric dipole (k = 1, Δℓ = ±1) oscillator strengths and line strengths can readily be derived for principal quantum numbers of several hundred, and, in addition, electric quadrupole (k = 2, Δℓ = 0, ± 2) and electric octupole (k = 3, Δℓ = ±1) matrix elements are derived and tabulated. While these results are ‘exact’ for non-relativistic, hydrogenic atoms and ions, they clearly provide useful approximate values for Rydberg states of any atomic radiator. As an illustration of the potential usefulness of these results, we refer to Griem's formulation of the Stark broadening for transitions between states of high principal quantum number in hydrogen (Rydberg–Rydberg transitions), as recently generalized by Watson, for application to radio astronomical spectra from the H II regions. The present (recurrence relation) method, which is also compared to some similar methods in the literature, is shown to be very convenient, in that customary simplifications of the atomic matrix elements for large n and n′ can be entirely avoided, with very little price of additional complexity in the treatment. The present results are therefore available to assist in any computations involving Rydberg–Rydberg transitions, e.g. the re-evaluation of Stark broadening theory for such highly excited states, should new astronomical data continue to show the need for such work.