9-j Symbol - Recoupling of Four Angular Momentum Vectors | Recoupling Angular Momentum Vectors

Recoupling of Four Angular Momentum Vectors

Coupling of two angular momenta and is the construction of simultaneous eigenfunctions of and, where, as explained in the article on Clebsch-Gordan coefficients.

Coupling of three angular momenta can be done in several ways, as explained in the article on Racah W-coefficients. Using the notation and techniques of that article, total angular momentum states that arise from coupling the angular momentum vectors, and may be written as

$| ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle.$

Alternatively, one may first couple and to and and to, before coupling and to :

$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle.$

Both sets of functions provide a complete, orthonormal basis for the space with dimension spanned by

$|j_1 m_1\rangle |j_2 m_2\rangle |j_4 m_4\rangle |j_5 m_5\rangle, \;\; m_1=-j_1,\ldots,j_1;\;\; m_2=-j_2,\ldots,j_2;\;\; m_4=-j_4,\ldots,j_4;\;\;m_5=-j_5,\ldots,j_5.$

Hence, the transformation between the two sets is unitary and the matrix elements of the transformation are given by the scalar products of the functions. As in the case of the Racah W-coefficients the matrix elements are independent of the total angular momentum projection quantum number :

$|((j_1j_4)j_7, (j_2j_5)j_8)j_9m_9\rangle = \sum_{j_3}\sum_{j6} | ((j_1j_2)j_3, (j_4j_5)j_6)j_9m_9\rangle \langle ( (j_1j_2)j_3,(j_4j_5)j_6)j_9 | ((j_1 j_4)j_7,(j_2j_5)j_8)j_9\rangle.$