Angular momentum and Heisenberg's correspondence principle: III. Rotation matrix elements

Abstract

Heisenberg's correspondence principle is applied to the matrix elements of the rotation operator; in this way, an approximation for the reduced rotation matrix elements dJM'M( theta ) in terms of Bessel functions is obtained. It is shown that two distinct approximate forms are necessary to give sufficient accuracy over the entire range 0 to pi of the angle theta if the approximation is to be of value. The two forms are most accurate for theta near 0 and pi respectively, deteriorating as 0= pi /2 is approached; however, they retain a surprising degree of accuracy over the full range, particularly when (M'-M) is small and J large, the case for which the exact expression is most complex. Taken in conjunction with the results of the previous papers in this series, the present work allows Bessel function approximations to be obtained for both Clebsch-Gordan and Racah coefficients; indications as to the likely accuracy of such approximations are given.