The Wigner–Eckart theorem and selection rules

A spherical tensor of rank $k$ consists of $2k+1$ operator components that transform among themselves like a spin-$k$ multiplet. A scalar has rank zero, and an ordinary vector has rank one. Tensor rank describes transformation behavior, not the number of spatial indices in an arbitrary notation. A Cartesian array must be decomposed into irreducible parts before a single-rank theorem can be applied. State labels can include additional quantum numbers, collectively denoted by $\alpha$, which rotations leave untouched.

The commutator with the z generator says that component $q$ changes a magnetic label by $q$. The raising and lowering commutators relate all components and force their matrix elements into the same angular pattern. For a vector, adopt $T^{(1)}_0=T_z$ and $T^{(1)}_{+1}=-(T_x+iT_y)/\sqrt2$, with the corresponding positive combination for the negative component. Declaring this convention prevents the signs of circular components from being silently reversed.

Use the convention in which a matrix element equals a Clebsch–Gordan coefficient times a reduced matrix element, with no additional square-root factor. Other references place a square root of a multiplet dimension in the formula; their reduced matrix elements therefore have a different normalization. Either convention predicts the same actual amplitudes when used consistently. The reduced factor retains the physical units of the tensor operator because the angular coefficient is dimensionless.

The theorem applies between states belonging to specified irreducible rotation representations and to an operator with the stated tensor transformation law. An isotropic Hamiltonian supplies natural energy states with these labels, but the algebraic theorem is about representations, not a numerical approximation to the interaction. If an external field mixes different angular-momentum sectors, its actual eigenstates must first be expanded into the relevant basis. In that situation, directly assigning one unmixed label to the state can produce false selection rules.

A Clebsch–Gordan coefficient vanishes unless the final magnetic label equals the initial label plus the component index, and the final angular momentum satisfies the triangle inequalities. For a vector operator, zero-to-zero is forbidden even though a superficial rule saying the change is at most one might suggest otherwise. There can also be zeros of a particular angular coefficient inside the triangle range. Checking the actual coefficient is stronger than checking the broad necessary conditions.

Additional symmetries impose independent constraints. Parity can kill an electric-dipole amplitude whose angular coefficient is nonzero. A radial integral can vanish through cancellation, and a probing field might have no component that drives the needed tensor index. Transition rates also depend on field strength, final-state density, and dynamical assumptions. Thus the theorem organizes amplitudes and relative strengths, while a complete spectroscopy prediction needs the interaction Hamiltonian and the physical preparation of the radiation.