= Wigner-Eckart theorem
{c}
The <Wigner-Eckart theorem> separates the dependence of a <spherical tensor operator> on <angular momentum> projections from its dynamical content. In one standard convention,
$$
\langle\lambda'j'm'|T_q^{(k)}|\lambda jm\rangle
=\frac{\langle jm;kq|j'm'\rangle}{\sqrt{2j'+1}}
\langle\lambda'j'\Vert T^{(k)}\Vert\lambda j\rangle.
$$
The <reduced matrix element> on the right is independent of $m,m',q$. The <Clebsch-Gordan coefficients> impose $m'=m+q$ and $|j-k|\leq j'\leq j+k$; additional <selection rules>, including <parity>, can make the <reduced matrix element> vanish.
To see why the factorization holds, combine the rank-$k$ transformation law of the operator with the spin-$j$ transformation law of the initial state. This gives an <intertwining operator> from the <tensor product representation> $j\otimes k$ to the final representation $j'$. The <Clebsch-Gordan decomposition> contains each allowed $j'$ once. Therefore the corresponding <intertwining operator> is unique up to an overall scalar, which is the <reduced matrix element> after the chosen normalization. Ratios within the same pair of multiplets depend only on <Clebsch-Gordan coefficients>, whenever the common <reduced matrix element> is nonzero.
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