Some classes of ‘nontrivial zeroes’ of angular momentum addition coefficients
Thomas A. Heim, Jürgen Hinze, A. R. P. Rau · Journal of Physics A Mathematical and Theoretical · 2009
Angular momentum coupling in quantum physics obeys obvious symmetries of rotation and reflection so that the Clebsch–Gordan (vector coupling) coefficients or Wigner 3 j -symbols describing the coupling vanish unless these symmetries are satisfied. However, it has long been observed that there are ‘accidental’ or ‘nontrivial’ zeroes of some coefficients even when the obvious symmetries are satisfied. Partial explanations and conjectures on the systematics of some of these zeroes have been advanced. We provide some more and propose as well ‘near zeroes’ which, while not exactly vanishing, are extremely small in magnitude. Connections are made to zeroes of Legendre and hypergeometric polynomials and to classical and semi-classical pictures for the addition of angular momenta. A convenient ordering scheme for 3 j 's that incorporates Regge symmetries also emerges. Further aspects of our analysis concern radial matrix elements of powers of r in a Coulomb potential that have analogous expressions to 3 j 's as a result of a non-compact O (2, 1) counterpart of the O (3) group symmetry of rotations. Some remarks are made on possible realization in actual physical systems.