A note on recursive calculations of particular 9jcoefficients

Abstract

The calculation of angular-momentum coupling transformation matrices can be very time consuming and alternative methods, even if they apply only in special cases, are helpful. We present a recursion relation for the calculation of particular 9j symbols used in the quantum theory of angular momentum.Keywords: angular momentum, 9j coefficient, recursion relation 03.65.Fd, The coupling of N angular momenta is related to the definition of a 3(N − 1)j symbol. The 9j symbols, characterizing the coupling of four angular momenta, are involved for instance in the computation of the matrix elements of the products of tensor operators, and in the transformation from LS to jj coupling [1,2]. The triple sum series of Jucys and Bandzaitis [3] is the simplest known algebraic form for the 9j coefficient. Fourty years ago, Ališauskas and Jucys [4][5][6][7][8] derived an algebraic expression, in which 9j symbols are written as the threefold summation of multiplications and divisions of factorials. More recently, Wei [9,10] proposed to express 9j symbols as a summation of the products of binomial coefficients and devised an algorithm to calculate the binomial coefficients recursively. Although some relationships between particular 9j symbols have been obtained in special cases [11], it is quite difficult to find recursion formulas for the 9j symbol itself, due to the fact that it contains nine arguments. PACSThe general recursion relations for arbitrary 9j symbols were introduced in [12], [3] (with incorrect phase factors), and in [13] (in corrected form). However, as a rule in these relations, several angular momentum parameters are changing. It seems that relations with changing single momentum parameter are possible only if some other parameters of the 9j accept extreme or fixed small values. For instance, the following 9j symbol: