Wigner-Eckart theorem

Let $𝑈 : 𝐺 \to G L (𝑉)$ be a unitary representation with its decomposition into irreps. Let ${O 𝜇 𝑖} 𝑑 𝜇 𝑖 = 1$ be irreducible operators transforming in $Γ 𝜇$ and ${| 𝑒 𝜈 𝛼 𝑗 ⟩} 𝑑 𝜈 𝑖 = 1$ be an irreducible orthonormal basis transforming in $Γ 𝜈$ . Following Irreducible operators applied to an irreducible orthonormal basis transform in the product representation, let $| 𝑖, 𝑗 ⟩$ denote $O 𝜇 𝑖 | 𝑒 𝜈 𝛼 𝑗 ⟩$ , and $| 𝑤 𝜆 𝛾 ℓ ⟩ = | 𝛾, 𝜆, ℓ ⟩$ denote the decomposed basis for the product. Then¹ rep

⟨ 𝑒 𝜆 𝛼 ℓ | O 𝜇 𝑖 | 𝑒 𝜈 𝛽 𝑗 ⟩ = \sum 𝛾 ⟨ 𝛾, 𝜆, ℓ (𝜇, 𝜈) 𝑖, 𝑗 ⟩ ⟨ 𝛼, 𝜆 ‖ O 𝜇 ‖ 𝛽, 𝜈 ⟩ 𝛾

where the so-called reduced matrix element is given by

⟨ 𝛼, 𝜆 ‖ O 𝜇 ‖ 𝛽, 𝜈 ⟩ 𝛾 = 1 𝑑 𝜆 \sum 𝑘 ⟨ 𝑒 𝜆 𝛼 𝑘 | 𝛾, 𝜆, 𝑘 ⟩

Proof

Both ${| 𝑒 𝜇 𝛼 𝑖 ⟩} 𝑑 𝜇 𝑖 = 1$ and ${| 𝛾, 𝜆, ℓ ⟩} 𝑑 𝜆 𝜆 = 1$ are Irreducible orthonormal basis vectors, but may not coïncide exactly for each irreducible invariant subspace, hence the reduced matrix element:
$⟨ 𝑒 𝜆 𝛼 ℓ | O 𝜇 𝑖 | 𝑒 𝜈 𝛽 𝑗 ⟩ = \sum 𝜌; 𝛾, 𝑚 ⟨ 𝑒 𝜆 𝛼 ℓ | 𝛾, 𝜌, 𝑚 ⟩ ⟨ 𝛾, 𝜌, 𝑚 | 𝑖, 𝑗 ⟩ = \sum 𝜌; 𝛾, 𝑚 ⟨ 𝛾, 𝜌, 𝑚 | 𝑖, 𝑗 ⟩ 1 𝑑 𝜆 \sum 𝑘 ⟨ 𝑒 𝜆 𝛼 𝑘 | 𝛾, 𝜆, 𝑘 ⟩ = \sum 𝑘 ⟨ 𝑒 𝜆 𝛼 𝑘 | 𝛾, 𝜆, 𝑘 ⟩ ⟨ 𝛼, 𝜆 ‖ O 𝜇 ‖ 𝛽, 𝜈 ⟩ 𝛾$
as required.

develop | en | SemBr

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

Wigner-Eckart theorem

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Wigner-Eckart theorem

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