D. Andrae, RECURSIVE EVALUATION OF EXPECTATION VALUES (R(K)) FOR ARBITRARY STATES OF THE RELATIVISTIC ONE-ELECTRON ATOM, Journal of physics. B, Atomic molecular and optical physics, 30(20), 1997, pp. 4435-4451
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.