The selection rules are special cases of one theorem. Operators, like states, come in families that rotate into one another: a scalar is unchanged by rotations, a vector’s three components mix like the three states of $j=1$, and a tensor operator of rank $k$ has $2k+1$ components that rotate like the states $\ket{k,q}$. The Wigner–Eckart theorem says that all the matrix elements of such an operator between two multiplets are one number, the reduced matrix element, times a Clebsch–Gordan coefficient. The geometry — which $m$, $m'$ and $q$ — is in the coefficient; the physics — how strong — is in the one number.

How bright a lamp looks depends on its power and on the direction you look from. The power is one number for the lamp; the direction gives a fixed geometric factor. The Wigner–Eckart theorem splits every matrix element of a tensor operator the same way.

For a tensor operator $\hat{T}^{(k)}_{q}$ of rank $k$, $q=-k$:

• A tensor operator of rank $k$ is defined by $[\hat{J}_{z},\hat{T}_{q}]=\hbar q\hat{T}_{q}$ and .

The commutators of a tensor operator with $\hat{\mathbf{J}}$ copy the way $\hat{J}_{\pm}$ act on the states $\mathrm{\mid }$. So the kets are raised and lowered exactly like the products , and can be combined with Clebsch–Gordan coefficients into kets that behave like . The overlap of such a ket with is zero unless and , and does not depend on . Undoing the combination leaves one Clebsch–Gordan coefficient times a number that knows nothing of , or .

The definition of a tensor operator of rank $k$.

So $\hat{T}_{q}\ket{j,m}$ is raised and lowered like the product $\ket{j,m}\ket{k,q}$.

Combined like a coupled state, it is raised and lowered like $\ket{J,M}$.

Kets that behave like $\ket{J,M}$ are orthogonal to other $J$ and $M$, and moving between values of $M$ with $\hat{J}_{\pm}$ shows the overlap is the same for each.

Undo the combination with the orthogonality of the coefficients: $\hat{T}_{q}\ket{j,m}=\sum_{J,M}\braket{j\,m;k\,q|J\,M}\ket{\Phi_{J,M}}$.

Every matrix element of a tensor operator between two multiplets is a Clebsch–Gordan coefficient, which holds all the dependence on $m$, $m'$ and $q$, times one reduced matrix element, which holds none.

A scalar is a tensor of rank 0. The Clebsch–Gordan coefficient for adding 0 to $j$ is zero unless $j'=j$ and $m'=m$.

The Clebsch–Gordan coefficient vanishes unless the projections add, so a component $\hat{T}_{q}$ moves $m$ by exactly $q$.

The three components of a vector operator, recombined, rotate like the states of $j=1$.

Great work! You now know how to:

• recognise scalar, vector and higher tensor operators by their commutators
• state the Wigner–Eckart theorem and derive its selection rules
• separate a matrix element into geometry and a reduced element