Wigner–Eckart Theorem

Statement

For a spherical tensor operator $T_q^{(k)}$, rotational symmetry factors angular-momentum matrix elements into a known angular coefficient and a reduced matrix element independent of magnetic quantum numbers.

With the convention used in the formula card,

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

Selection Rules

The angular coefficient enforces

$$m'=m+q, \qquad \lvert j-k\rvert\le j'\le j+k.$$

Other symmetries, such as parity or exchange symmetry, can add further selection rules.

Assumptions

• The states transform as angular-momentum eigenstates under the same $SU(2)$ convention.
• $T_q^{(k)}$ is a spherical tensor operator of rank $k$.
• Clebsch-Gordan and Wigner $3j$ phase conventions are used consistently.
• The reduced-matrix-element normalization matches the chosen convention.

Quantum-Mechanical Meaning

The theorem says that symmetry controls the dependence on $m,m',q$. Dynamics and radial or internal structure are contained in the reduced matrix element. This is why large families of transition amplitudes can be related before doing detailed integrals.

Canonical Links

• Wigner–Eckart Theorem Explained
• Wigner–Eckart Formula Card
• Clebsch-Gordan Coefficients
• Wigner 3j, 6j, and 9j Symbols
• Angular Momentum Algebra
• Why Symmetry Matters

Common Mistakes

• Mixing reduced-matrix-element conventions from different books.
• Treating the reduced matrix element as fixed by symmetry alone.
• Forgetting that parity and other discrete symmetries supply additional rules.
• Applying the theorem to an operator that is not a spherical tensor.
• Losing phase signs by switching between $3j$ and Clebsch-Gordan conventions.

Quick Check

What does the theorem say for a scalar operator with $k=0$?

Solution

Only $q=0$ exists. The selection rules give $m'=m$ and $j'=j$. A rotational scalar is diagonal in $j$ and $m$, up to any additional degeneracy labels not shown.

References

• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.