Quantum numbers and Angular Momentum Algebra

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Angular momentum algebra, Wigner-Eckart theorem

With the

$6j$ symbol defined, we can go back and and rewrite the overlap between the two ways of recoupling angular momenta in terms of the

$6j$ symbol. That is, we can have

$\vert (j_a\rightarrow [j_b\rightarrow j_c]J_{bc}) JM\rangle =\sum_{J_{ab}}(-1)^{j_a+j_b+j_c+J}\sqrt{(2J_{ab}+1)(2J_{bc}+1)}\left\{\begin{array}{ccc} j_a & j_b& J_{ab} \\ j_c & J & J_{bc} \end{array}\right\}| ([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) JM\rangle.$ Can you find the inverse relation? These relations can in turn be used to write out the fully anti-symmetrized three-body wave function in a

$J$ -scheme coupled basis. If you opt then for a specific coupling order, say

$| ([j_a\rightarrow j_b]J_{ab}\rightarrow j_c) JM\rangle$ , you need to express this representation in terms of the other coupling possibilities.