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Angular momentum algebra, Wigner-Eckart theorem

We can also coupled together four angular momenta. Consider two four-body states, with single-particle angular momenta j_a , j_b , j_c and j_d we can have a state with final J |\Phi(a,b,c,d)\rangle_1 = | ([j_a\rightarrow j_b]J_{ab}\times [j_c\rightarrow j_d]J_{cd}) JM\rangle, where we read the coupling order as j_a couples with j_b to given and intermediate angular momentum J_{ab} . Moreover, j_c couples with j_d to given and intermediate angular momentum J_{cd} . The two intermediate angular momenta J_{ab} and J_{cd} are in turn coupled to a final J . These operations involved three Clebsch-Gordan coefficients.

Alternatively, we could couple in the following order |\Phi(a,b,c,d)\rangle_2 = | ([j_a\rightarrow j_c]J_{ac}\times [j_b\rightarrow j_d]J_{bd}) JM\rangle,