Journal of Physics A: Mathematical and General
The last in a series of three papers in which it is shown how the radial part of non-relativistic and relativistic hydrogenic bound-state calculations involving the Green functions can be presented in a unified manner. The work presented here is concerned with the reduced Green functions which arise in second-order stationary state perturbation theory. Using a simple linear transformation of the four radial parts of the relativistic reduced Green function it is shown how the non-relativistic and relativistic functions are special instances of the solution of a general second-order differential equation. The general solution of this equation is exhibited in the form of a Sturmian expansion, and complete solutions in both cases are presented. Recursion relations are deduced for the radial parts of both reduced Green functions and their matrix elements are examined in detail. As a test of the given functions the second-order effect of a perturbation of the nuclear charge is calculated and is shown to agree exactly with the value expected from a simple Taylor expansion of the hydrogenic energy formula.
Swainson, R. A. and Drake, Gordon W. F.. (1991). A unified treatment of the non-relativistic and relativistic hydrogen atom: III. The reduced Green functions. Journal of Physics A: Mathematical and General, 24 (8), 1801-1824.