Representation operator

Irreducible operators applied to an irreducible orthonormal basis

Let be a unitary representation with its decomposition into irreps. Let be irreducible operators transforming in and be an irreducible orthonormal basis transforming in . Then transform under in the product representation (within the same carrier space)

which may then be expanded in a decomposed reälization using the Clebsch-Gordan coëfficients.

giving rise to the Wigner-Eckart theorem.

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