Wigner–Eckart Theorem

The Wigner–Eckart theorem is the workhorse theorem behind angular-momentum selection rules. It says that, for an operator with definite rotational transformation law, every matrix element splits into two parts:

• a universal angular coefficient fixed by symmetry;
• a reduced matrix element containing the dynamics, radial integrals, charges, coupling constants, and internal structure.

Thus many transition amplitudes can be related before any explicit wavefunction integral is done.

Matrix Elements Between Angular-Momentum Multiplets

Suppose states are labeled by angular momentum:

$$J^2\lvert\alpha,j,m\rangle = \hbar^2j(j+1)\lvert\alpha,j,m\rangle, \qquad J_z\lvert\alpha,j,m\rangle = \hbar m\lvert\alpha,j,m\rangle.$$

The label $\alpha$ stands for all additional quantum numbers not displayed explicitly: radial labels, parity, principal quantum number, spin-coupling labels, particle species, or degeneracy labels.

A generic operator $O$ has unrelated matrix elements

$$\langle\alpha',j',m'|O|\alpha,j,m\rangle.$$

If $O$ is a spherical tensor component $T_q^{(k)}$, rotational symmetry makes those matrix elements highly structured. The dependence on $m,m'$, and $q$ is not a new dynamical calculation each time; it is fixed by Clebsch–Gordan algebra.

Spherical Tensor Input

An irreducible spherical tensor operator of rank $k$ has components

$$T_q^{(k)}, \qquad q=-k,-k+1,\ldots,k.$$

These components transform under rotations like a spin-$k$ multiplet. Infinitesimally, this is encoded by

$$[J_z,T_q^{(k)}] = \hbar q\,T_q^{(k)},$$

and

$$[J_\pm,T_q^{(k)}] = \hbar \sqrt{k(k+1)-q(q\pm1)} \,T_{q\pm1}^{(k)}.$$

Scalars have $k=0$. Vector operators, such as position $\mathbf R$ or dipole moment components, have $k=1$ after conversion to spherical components. Quadrupole operators are typical rank-$2$ tensors.

Statement

We use the following convention for the reduced matrix element:

$$\langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle = (-1)^{j'-m'} \begin{pmatrix} j'&k&j\\ -m'&q&m \end{pmatrix} \langle\alpha',j'\lVert T^{(k)}\rVert\alpha,j\rangle.$$

Equivalently, using the matching Clebsch–Gordan convention,

$$\langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle = \frac{ \langle j,m;k,q|j',m'\rangle }{ \sqrt{2j'+1} } \langle\alpha',j'\lVert T^{(k)}\rVert\alpha,j\rangle.$$

The double bars denote a reduced matrix element. Different books place factors such as $\sqrt{2j+1}$ or $\sqrt{2j'+1}$ differently, so this convention must be checked before comparing tables.

Selection Rules

The angular coefficient vanishes unless the magnetic quantum numbers add consistently:

$$m'=m+q.$$

It also vanishes unless the angular momenta satisfy the triangle condition:

$$|j-k| \le j' \le j+k.$$

There are also the ordinary validity conditions

$$|m|\le j, \qquad |m'|\le j', \qquad |q|\le k,$$

together with $j+j'+k\in\mathbb Z$. Equivalently, the allowed $j'$ values run from $|j-k|$ to $j+k$ in integer steps.

These are rotational selection rules. They do not include parity, charge conjugation, exchange symmetry, isospin, molecular point-group symmetry, or any dynamical zero of the reduced matrix element.

Meaning of the Reduced Matrix Element

The reduced matrix element

$$\langle\alpha',j'\lVert T^{(k)}\rVert\alpha,j\rangle$$

is independent of $m,m'$, and $q$. It can still depend on:

• the operator $T^{(k)}$ itself;
• the initial and final total angular momenta $j,j'$;
• the additional state labels $\alpha,\alpha'$;
• radial wavefunctions, coupling constants, effective charges, and model assumptions;
• the chosen reduced-matrix-element convention.

The theorem does not compute this number. It says that once this number is known for a pair of multiplets and an operator, all magnetic-sublevel matrix elements follow by angular algebra.

Symmetry Interpretation

The operator component $T_q^{(k)}$ carries angular momentum $k$ and magnetic label $q$. Acting on a state in the $j$ multiplet, it produces an object transforming like the tensor product

$$j\otimes k.$$

The final state can overlap with this object only if the $j'$ representation appears in that tensor product. This is the triangle condition.

The magnetic label must also match:

$$m+q=m'.$$

Each $j'$ occurs once in the SU(2) product $j\otimes k$. Therefore the map from that product into a specified final multiplet is, up to one number, the same map supplied by the Clebsch–Gordan coefficients. This multiplicity-one fact explains why there is just one reduced matrix element for each pair of specified multiplets. Extra labels $\alpha,\alpha'$ distinguish different copies; they do not have to share the same reduced number.

The theorem is about transformation laws and does not itself require $[H,\mathbf J]=0$. Rotational invariance of $H$ is what lets its energy eigenstates be organized into the $j,m$ multiplets used in a simple spectroscopy application.

Scalar Operators

For a rotational scalar, $k=0$ and $q=0$. The selection rules become

$$m'=m, \qquad j'=j.$$

Thus a scalar cannot change $j$ or $m$. No nondegeneracy assumption is needed. It may still act nontrivially on additional labels:

$$\langle\alpha',j,m| T^{(0)}_0 |\alpha,j,m\rangle$$

can depend on $\alpha$ and $\alpha'$. In our normalization the full formula is

$$\langle\alpha',j',m'|T^{(0)}_0|\alpha,j,m\rangle =\delta_{j'j}\delta_{m'm} \frac{\langle\alpha',j\lVert T^{(0)}\rVert\alpha,j\rangle}{\sqrt{2j+1}}.$$

The remaining matrix in $\alpha',\alpha$ can mix radial states or repeated copies of the same $j$ representation. It is identical for every $m$ in that multiplet.

Vector Operators and Dipole Rules

A vector operator is a rank-$1$ spherical tensor. Its components have

$$k=1, \qquad q=-1,0,1.$$

The Wigner–Eckart theorem gives

$$m'=m+q,$$

and

$$|j-1| \le j' \le j+1.$$

For orbital angular momentum states, the electric dipole operator is proportional to the position vector, so the rotational rule allows

$$\ell'=\ell-1,\ell,\ell+1$$

subject to the triangle condition. But the position operator is odd under parity. Hydrogenic orbital parity is $(-1)^\ell$, so parity selection rules require $\ell'$ and $\ell$ to have opposite parity. Combining the rotational and parity rules gives the familiar electric-dipole orbital rule

$$\Delta\ell=\pm1.$$

The magnetic rule remains

$$\Delta m=q,$$

where $q$ is fixed by the polarization component of the driving field.

Worked Example

Consider a rank-$1$ operator connecting states with $j=1/2$ to final states with possible $j'$. The triangle rule gives

$$\left| \frac12-1 \right| \le j' \le \frac12+1.$$

Therefore

$$j'=\frac12 \quad \text{or} \quad j'=\frac32.$$

For the component $q=+1$, the magnetic rule gives

$$m'=m+1.$$

If the initial state has $m=1/2$, then $m'=3/2$. This immediately rules out the $j'=1/2$ final multiplet for that component, because $m'=3/2$ is outside the allowed range for $j'=1/2$. The only possible final angular state is $j'=3/2,m'=3/2$, provided no additional symmetry or reduced-matrix-element zero forbids it.

What the Theorem Does Not Say

The Wigner–Eckart theorem is powerful because it is kinematic. That is also its limitation.

It does not say:

• what the energy levels are;
• what the reduced matrix element is numerically;
• whether the radial integral vanishes accidentally;
• whether parity or another discrete symmetry forbids the transition;
• whether a weakly forbidden transition can occur through a different operator or higher-order process;
• which coupling scheme is physically appropriate in a strong external field.

It is a theorem about rotational covariance, not a complete spectroscopy model.

Common Mistakes

• Treating the reduced matrix element as a universal constant. It depends on the states and on the physical operator.
• Mixing $3j$ and Clebsch–Gordan conventions from different sources.
• Forgetting the $m'=m+q$ rule.
• Applying angular-momentum selection rules but forgetting parity.
• Assuming that allowed by symmetry means numerically large. Allowed matrix elements can still be small or zero for dynamical reasons.
• Using the theorem for an operator before showing that it transforms as a spherical tensor.

Exercises

1. Find the reduced matrix element of $\mathbf J$ inside one spin-$j$ multiplet. Use $J_0=J_z$ and, for $j>0$, $\langle j,m;1,0|j,m\rangle=m/\sqrt{j(j+1)}$.

Solution

Choose any $m\ne0$. The directly known matrix element is $\hbar m$. Inserting the supplied coefficient gives

$$\hbar m=\frac{m}{\sqrt{j(j+1)(2j+1)}} \langle j\lVert J^{(1)}\rVert j\rangle,$$

and hence

$$\langle j\lVert J^{(1)}\rVert j\rangle =\hbar\sqrt{j(j+1)(2j+1)}.$$

For $j=0$ every component of $\mathbf J$ vanishes and the reduced matrix element is zero as well. The result provides a normalization check when comparing another book’s convention.

2. For a scalar operator, show that $j'=j$ and $m'=m$.

Solution

A scalar has $k=0$ and $q=0$. The magnetic rule gives

$$m'=m.$$

The triangle rule gives

$$|j-0|\le j'\le j+0,$$

so

$$j'=j.$$

3. A vector operator acts on an initial state with $j=0,m=0$. Which final $j',m'$ values are allowed by rotations?

Solution

For a vector operator, $k=1$ and $q=-1,0,1$. The triangle rule gives

$$|0-1|\le j'\le0+1,$$

so

$$j'=1.$$

The magnetic rule gives

$$m'=0+q=q.$$

Thus the allowed final magnetic labels are

$$m'=-1,0,1,$$

all inside the $j'=1$ multiplet.

4. Explain why the electric dipole rule $\Delta\ell=\pm1$ is not just the triangle rule.

Solution

The electric dipole operator is a vector, so angular momentum alone allows

$$\ell'=\ell-1,\ell,\ell+1$$

where valid. However, the dipole operator is odd under parity. Orbital states have parity $(-1)^\ell$, so a nonzero dipole matrix element requires opposite parity:

$$(-1)^{\ell'}=-(-1)^\ell.$$

Thus $\ell'$ and $\ell$ must differ by an odd integer. Combining this with the vector triangle rule leaves

$$\Delta\ell=\pm1.$$

References

• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.