Clebsch-Gordon expansion

YkM is a vector coupled product of solid harmonics[69] given by the Clebsch-Gordon expansion,... 

The kappa values are readily evaluated using the Clebsch-Gordon expansion [7.14] and are given below in terms of D q = (P2) and D q = (P4), the ensemble average of the fourth-rank Legendre polynomial. 

In this example, there are nine uncoupled states and, of course, nine coupled states. The coefficients that have been worked out are linear expansion coefficients, and so they can be arranged in a 9 x 9 matrix. This is the transformation (matrix) from the uncoupled basis to the coupled basis. This matrix and its inverse contain the coefficients for making up one kind of basis function from the other set. These coefficients are used in many problems in quantum physics, and they are often called vector coupling coefficients. Since they can be worked out once and for all for specific choices of /j and there are many tabulations of the values, and many alternate procedures have been devised for finding them. They are sometimes scaled by different factors for convenience in different circumstances. Names given to vector coupling coefficients under various prescriptions include Clebsch-Gordon coefficients, Racah coefficients, 3 -j symbols, and 6- j symbols. 

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