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The Wigner-Eckart theorem separates the dependence of a spherical tensor operator on angular momentum projections from its dynamical content. In one standard convention,The reduced matrix element on the right is independent of $m,m^{\prime },q$. The Clebsch-Gordan coefficients impose $m^{\prime }=m+q$ and $|j-k|\le j^{\prime }\le j+k$; 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 $j^{\prime }$. The Clebsch-Gordan decomposition contains each allowed $j^{\prime }$ 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.

A Zeeman sublevel is a magnetic-field-resolved component of an angular-momentum level, labeled by its projection quantum number $m$. In the weak-field regime, $\Delta E=g{\mu }_{B}mB$ for the relevant electronic angular momentum, with appropriate modifications for hyperfine levels.

For angular momentum one, the values $m=-1,0,1$ give three shifts. The states can be degenerate at zero field yet acquire different energies in a field. Light polarization and selection rules control which sublevels a laser addresses. Strong fields can change the angular-momentum coupling, so one cannot extrapolate every weak-field formula indefinitely.