Representation operator

Wigner-Eckart theorem

Let be a unitary representation with its decomposition into irreps. Let be irreducible operators transforming in and be an irreducible orthonormal basis transforming in . Following Irreducible operators applied to an irreducible orthonormal basis transform in the product representation, let denote , and denote the decomposed basis for the product. Then¹ rep

where the so-called reduced matrix element is given by

Proof

Both and are Irreducible orthonormal basis vectors, but may not coïncide exactly for each irreducible invariant subspace, hence the reduced matrix element:

as required.

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Footnotes

1. 2023, Groups and representations, §4.2, pp. 54–55 ↩