Matrix Elements and the Wigner-Eckart Theorem

A general interaction element is a bracket around an operator. Each of the three ingredients, bra, ket, and operator, can be put in symmetry-adapted form, so that it transforms according to a given irrep. Moreover, provided that the symmetry adaptation is done properly, not only the irrep itself but also the subrepresentation is well defined. Altogether, the matrix element will thus be characterized by six symmetry labels, as The labels imply that the symmetry behaviour of each of 

Note that the general form of the operator [ refers to a component of an ir- 

Theorem 12 A matrix element, involving a tensor operator, may be factorized into a product of an intrinsic scalar part and an appropriate 3F coupling coefficient. 

To prove this theorem, one first considers the coupling of two ingredients of the matrix element, and then compares the result with the third one. We thus first consider the coupling of the operator and the ket. The transformation of their product does indeed correspond to the super matrix which is due to the direct product Ay . r. 

We now invert this equation, using the unitary properties of the coupling coefficients, to yield  

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