Physica Scripta. Vol. 51, 9-12, 1995 zyxwvuts Matrix Elements zyxwv (n Irfl Is) for Arbitrary zyx p with Relativistic Hydrogenic Functions Yousef I. Salamin Physics Department, Birzeit University, P.O. Box 14 zyxwvutsrqp - Birzeit, West Bank, via Israel zyxwvuts Received April 14,1994; accepted in revised zyxwvutsrq form June 8, 1994 zyxwvutsrq Abstract We report an analytic expression by means of which generalized matrix elements of r), for arbitrary and using the relativistic Dirac-Coulomb wavefunctions, may be calculated exactly. A couple of examples are worked out fully and are shown to reproduce their nonrelativistic counterparts, available in the literature, when the nonrelativistic quantum numbers are used. 1. Introduction Analytic evaluation of radial matrix elements using hydrog- enic wavefunctions has been the subject of many papers [l-31, some of them are as old as quantum mechanics itself [4]. Those matrix elements are encountered in a wide range of quantum mechanical calculations involving Rydberg atoms [5, 61. The analytic forms are sought after because they lead to precise theoretical results for comparison with experimental values. Most efforts, however, have been devoted to evaluating matrix elements of the form (nl I r8 I n'r) of integer values of the exponent and using nonrelativistic hydrogenic wave- functions. r here is the radial coordinate. Drake and Swain- son [ 11 have recently reported nonrelativistic analytic expectation values of rs for arbitrary hydrogenic states. More recently, Shertzer [2] has reported simple nonrelativ- istic analytic expressions for matrix elements of this form for arbitrary values of zyxwvutsrqp p, relevant for some calculations with hydrogen and helium Rydberg electrons. The relativistic calculation of similar matrix elements has enjoyed less attention over the years. In 1962, Garstang and Mayers [7] calculated matrix elements of this kind of p = 1 and 2. Later, in 1967, Burke and Grant [SI performed similar calculations for p = -3, - 1, 1 and 2. In this paper, the effort is devoted to a calculation of matrix elements of the form (n I r@ I s), for p not necessarily restricted to integer values, using relativistic (Dirac- Coulomb) wavefunctions. In the notation employed here, n and s stand for all the quantum numbers of the terms nL, and sL, (In) InJIM)). A calculation of this nature may turn out to be important for some quantum mechanical applications where hydrogenic atoms of high Z values are involved, in which regime the relativistic corrections become more pronounced. These elements show up also in the asymptotic expressions encountered in the study of many properties pertaining to Rydberg atoms. The author's main interest in this work, however, is mathematical, as reduction of the generalized radial integrals, in a fashion to be present- ed below, saves a great deal of time and effort which, other- wise, one spends multiplying term by term, integrating and summing lengthy series. 2. Background In order to make this paper as self-contained as possible, we devote the present section to a quick review of the Dirac- Coulomb wavefunctions. We employ the Lorentz-Heaviside system of units h = c = 1. In this system of units, the rela- tivistic one-electron (Dirac) equation, for a Coulomb field U(r) = -Zct/r, has the (bound state) solutions In eq. (l), the spherical spinors contain all the angular and spin dependences and satisfy the normalization condi- tion F The radial dependence, on the other hand, is entirely in [9] where the upper (lower) sign goes with g(f). Also (23~~'~ [ r(2y + n, + 1)11/' U= r(2y + 1) 4N(N - ~)n,! ' (4) A(r) = nrlFl(-nr + 1, 2y + 1; 2Ir)(213r)Y-'e-", B(r) = (N - K)~F~(-~,, 27 + 1; 21r)Y-1e-'r. (5) (6) Other quantities encountered so far have the following defi- nitions I = Zam/N, N = Jn' - 2n,( I K I - y), E,, = J-, (7) y = JK' - (Za)', n,=n-IKJ, and finally e if J = e - zy 4, -(L+l) ifJ=t'+i. K={ (8) Physica Scripta 51