An alternative approach to calculating the mean values rs for relativistic hydrogen-like atoms

A direct approach for deriving the formulas of the mean values <r)supers bar>s of hydrogen-like atoms for an integer s is developed in relativistic quantum mechanics. Explicit formulas of<(r)supers bar>s for \s\ less than or equal to 6 are presented and their non-relativistic limits are investigated resulting in explicit formulas of r(s) for hydrogen-like atoms in non-relativistic quantum mechanics. The forward and backward two-step uncoupled recursion relations of 7 are also constructed out. Formulas of r(s) for \s\less than or equal to 6 are demonstrated satisfying both two-step uncoupled recursion relations and one-step coupled recursion relations. (C) 2004 Elsevier B.V. All rights reserved.