Generalization of Stevens' operator-equivalent method

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Abstract
The operator-equivalent method was introduced by Stevens in 1952. This method enabled him to determine the quantum mechanical equivalent of a given spherical harmonic Ckq-the so-called Racah tensor operator Ckq-as an explicit function of the total angular momentum operator J within a constant J manifold. The method itself uses spherical harmonics in Cartesian coordinates and from each spherical harmonic the corresponding Racah tensor operator is calculated. This paper shows that it is useful to fix tau k- mod q mod as all spherical harmonics Cmod q mod + tau , mod q mod possess the same polynomial structure with coefficients showing a simple dependence on the absolute value of q. For the following Racah operators Cmod q mod + tau , mod q mod , which hold for all mod q mod , Stevens' operator-equivalent method has to be generalized. By an application of the Wigner-Eckart theorem to the operator-equivalents Cmod q mod + tau ,mod q, mod (J), the 3j-symbols with two identical js disclose its innermost functional dependence on the variables. To give an example the calculations are explicitly stated for tau =0, 1, 2, . . .5.

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