Recursive evaluation of expectation values (r(k)) for arbitrary states of the relativistic one-electron atom

Andrae D (1997)
JOURNAL OF PHYSICS B-ATOMIC MOLECULAR AND OPTICAL PHYSICS 30(20): 4435-4451.

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Abstract
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 approaches, the present algorithm is based on only a single recurrence relation, which is a recurrence relation for a special type of generalized hypergeometric F-3(2) series. Two sequences of values of such 3 F-2 series are generated. Finally, the members of these two sequences are assembled to yield the expectation values. The algorithm can be considered as a relativistic analogue of the Kramers-Pasternack recursion for expectation values (r(k)) for states of the non-relativistic hydrogen-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 addition, numerical values for (r(k)) are given for some representative states of one-electron atoms with Z = 1, 80 and 137.
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Andrae D. 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. 1997;30(20):4435-4451.
Andrae, D. (1997). 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), 4435-4451.
Andrae, D. (1997). 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, 4435-4451.
Andrae, D., 1997. 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), p 4435-4451.
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, vol. 30, 1997, pp. 4435-4451.
Andrae, D.: 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, 4435-4451 (1997).
Andrae, Dirk. “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): 4435-4451.
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