RECURSIVE EVALUATION OF EXPECTATION VALUES (R(K)) FOR ARBITRARY STATES OF THE RELATIVISTIC ONE-ELECTRON ATOM

An algorithm for the recursive evaluation of expectation values (r(k)) for a given state of the relativistic one-electron atom with a point- like nucleus of charge number Z is proposed. In contrast to previous a pproaches, the present algorithm is based on only a single recurrence relation, which is a recurrence relation for a special type of general ized hypergeometric F-3(2) series. Two sequences of values of such 3 F -2 series are generated. Finally, the members of these two sequences a re assembled to yield the expectation values. The algorithm can be con sidered as a relativistic analogue of the Kramers-Pasternack recursion for expectation values (r(k)) for states of the non-relativistic hydr ogen-like atom. As an application closed-form expressions for (r(k)), -6 less than or equal to k less than or equal to 5, were derived. In a ddition, numerical values for (r(k)) are given for some representative states of one-electron atoms with Z = 1, 80 and 137.