Wigner 3j, 6j, and 9j Symbols

Wigner symbols are compact invariant coefficients for angular momentum coupling. They package the bookkeeping of $SU(2)$ representation theory into quantities with strong symmetry properties.

The three most common symbols answer three different questions:

• A $3j$ symbol is a phase-normalized version of a Clebsch–Gordan coefficient.
• A $6j$ symbol changes the coupling order of three angular momenta.
• A $9j$ symbol changes the coupling scheme of four angular momenta.

They are convention-sensitive. This page uses the same Condon–Shortley convention as Angular Momentum Algebra and Wigner D-Matrices.

Three-j Symbols

The Wigner $3j$ symbol is written

$$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}.$$

It is nonzero only if the magnetic quantum numbers sum to zero:

$$m_1+m_2+m_3=0.$$

It is also nonzero only when $j_1$, $j_2$, and $j_3$ satisfy the triangle conditions:

$$\lvert j_1-j_2\rvert \leq j_3 \leq j_1+j_2,$$

together with the two cyclic variants. In the usual angular momentum setting, $j_1+j_2+j_3$ must be an integer and each $\lvert m_i\rvert\leq j_i$.

The relation to Clebsch–Gordan coefficients is

$$\langle j_1,m_1;j_2,m_2\vert J,M\rangle = (-1)^{j_1-j_2+M} \sqrt{2J+1} \begin{pmatrix} j_1 & j_2 & J\\ m_1 & m_2 & -M \end{pmatrix}.$$

Equivalently,

$$\begin{pmatrix} j_1 & j_2 & J\\ m_1 & m_2 & -M \end{pmatrix} = \frac{(-1)^{j_1-j_2+M}}{\sqrt{2J+1}} \langle j_1,m_1;j_2,m_2\vert J,M\rangle.$$

Thus a $3j$ table and a Clebsch–Gordan table contain the same information, but not in the same normalization or phase layout.

Why Three-j Symbols Are Useful

The $3j$ notation makes symmetries more visible than ordinary Clebsch–Gordan notation. Under an odd permutation of columns,

$$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_2 & j_1 & j_3\\ m_2 & m_1 & m_3 \end{pmatrix}.$$

Under simultaneous sign reversal of all magnetic labels,

$$\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix} = (-1)^{j_1+j_2+j_3} \begin{pmatrix} j_1 & j_2 & j_3\\ -m_1 & -m_2 & -m_3 \end{pmatrix}.$$

Even permutations leave the symbol unchanged. These symmetry properties are why angular integrals, spherical tensor matrix elements, and selection-rule calculations are often written with $3j$ symbols.

Orthogonality follows from Clebsch–Gordan unitarity. With the conventions above,

$$\sum_{m_1,m_2} (2J+1) \begin{pmatrix} j_1 & j_2 & J\\ m_1 & m_2 & -M \end{pmatrix} \begin{pmatrix} j_1 & j_2 & J'\\ m_1 & m_2 & -M' \end{pmatrix} = \delta_{JJ'}\delta_{MM'}.$$

Six-j Symbols

When three angular momenta are coupled, there is more than one natural order. One may first couple $j_1$ and $j_2$ to $j_{12}$, then couple the result with $j_3$ to total $J$. Or one may first couple $j_2$ and $j_3$ to $j_{23}$, then couple $j_1$ with that result.

The two coupled bases are related by a Wigner $6j$ symbol:

$$\begin{aligned} \lvert ((j_1j_2)j_{12}j_3)JM\rangle &= \sum_{j_{23}} (-1)^{j_1+j_2+j_3+J} \sqrt{(2j_{12}+1)(2j_{23}+1)} \\ &\quad\times \left\{ \begin{matrix} j_1 & j_2 & j_{12}\\ j_3 & J & j_{23} \end{matrix} \right\} \lvert (j_1(j_2j_3)j_{23})JM\rangle. \end{aligned}$$

The $6j$ symbol is therefore a recoupling coefficient. It does not depend on $M$, because it compares two ways of organizing the same irreducible total-$J$ space.

The nonzero conditions are compactly described by four triangle constraints. In the arrangement

$$\left\{ \begin{matrix} j_1 & j_2 & j_{12}\\ j_3 & J & j_{23} \end{matrix} \right\},$$

the relevant triples are

$$(j_1,j_2,j_{12}),\quad (j_1,J,j_{23}),\quad (j_3,j_2,j_{23}),\quad (j_3,J,j_{12}).$$

Each triple must be an allowed angular momentum coupling.

Nine-j Symbols

Four angular momenta can be paired in different ways. For example, compare the two schemes

$$((j_1j_2)j_{12}(j_3j_4)j_{34})J$$

and

$$((j_1j_3)j_{13}(j_2j_4)j_{24})J.$$

Their overlap is written with a Wigner $9j$ symbol:

$$\begin{aligned} &\left\langle ((j_1j_2)j_{12}(j_3j_4)j_{34})JM \vert ((j_1j_3)j_{13}(j_2j_4)j_{24})JM \right\rangle \\ &\quad= \sqrt{ (2j_{12}+1)(2j_{34}+1) (2j_{13}+1)(2j_{24}+1) } \left\{ \begin{matrix} j_1 & j_2 & j_{12}\\ j_3 & j_4 & j_{34}\\ j_{13} & j_{24} & J \end{matrix} \right\}. \end{aligned}$$

The rows and columns of the $9j$ array encode allowed angular momentum couplings. A $9j$ symbol is typically used when two different pairings of four angular momenta must be compared.

What Each Symbol Encodes

The three symbols form a hierarchy of basis transformations:

• $3j$ symbols encode one binary coupling and are equivalent to Clebsch–Gordan coefficients.
• $6j$ symbols encode reassociation of three angular momenta.
• $9j$ symbols encode a change of pairing for four angular momenta.

This hierarchy is not about stronger interactions or higher-order perturbation theory. It is about changing basis inside tensor products of irreducible angular momentum representations.

Selection Rules and Zeros

Wigner symbols are often useful because they vanish for structural reasons before any dynamics is considered.

For $3j$ symbols:

• $m_1+m_2+m_3=0$;
• each $\lvert m_i\rvert\leq j_i$;
• the three $j_i$ labels satisfy triangle inequalities;
• $j_1+j_2+j_3$ is an integer.

For $6j$ symbols, the four triangular triples must be allowed. For $9j$ symbols, each row and column must represent an allowed triangular coupling. These are necessary conditions; some additional cancellations can still occur because of phase and summation structure.

Applications

Wigner symbols appear in:

• addition of angular momentum beyond two spins;
• matrix elements of spherical tensor operators;
• atomic and molecular spectroscopy;
• partial-wave and multipole calculations;
• spin-network and recoupling algebra;
• changing between different coupling schemes in many-body bases;
• reduced matrix element calculations using the Wigner–Eckart theorem.

The symbols are mathematical coefficients. Dynamics enters through Hamiltonians, perturbations, potentials, and matrix elements; Wigner symbols only organize the angular momentum part.

Common Mistakes

• Copying a $3j$ symbol where a Clebsch–Gordan coefficient is required.
• Ignoring the phase convention used by a table.
• Forgetting the square-root dimension factors in $6j$ and $9j$ recoupling formulas.
• Treating a necessary triangle rule as a complete nonzero criterion.
• Assuming $6j$ and $9j$ symbols depend on the magnetic label $M$.
• Mixing active and passive rotation conventions when using Wigner D-matrices and Wigner symbols together.

Cross-Links

• Angular Momentum Algebra
• Tensor Product Representations
• Clebsch–Gordan Coefficients
• Wigner D-Matrices
• SU(2)
• Ladder Operators as Lie Algebra Tools
• Clebsch–Gordan Coefficients: Angular Momentum Derivation
• Selection Rules and Transition Rates

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.
• E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.

Exercises

1. Explain why

$$\begin{pmatrix} 1 & 1 & 1\\ 1 & 0 & 1 \end{pmatrix}$$

must vanish.

Solution

The lower labels must sum to zero. Here

$$1+0+1=2,$$

so the symbol vanishes before any detailed calculation is needed.

2. Convert the coefficient

$$\langle \tfrac12,\tfrac12;\tfrac12,-\tfrac12\vert 1,0\rangle = \frac{1}{\sqrt2}$$

into a $3j$ symbol.

Solution

Use

$$\langle j_1,m_1;j_2,m_2\vert J,M\rangle = (-1)^{j_1-j_2+M} \sqrt{2J+1} \begin{pmatrix} j_1 & j_2 & J\\ m_1 & m_2 & -M \end{pmatrix}.$$

Here $j_1=j_2=1/2$ and $M=0$, so the phase is $1$ and $\sqrt{2J+1}=\sqrt3$. Hence

$$\begin{pmatrix} \tfrac12 & \tfrac12 & 1\\ \tfrac12 & -\tfrac12 & 0 \end{pmatrix} = \frac{1}{\sqrt6}.$$

3. In a $6j$ symbol, why does the recoupling coefficient not depend on $M$?

Solution

Both coupling schemes describe the same total spin-$J$ irreducible representation. The change from $j_{12}$ to $j_{23}$ reorganizes multiplicity labels inside that total-$J$ sector; it does not act on the magnetic subspace within the irreducible representation. Rotational covariance therefore makes the recoupling coefficient independent of $M$.

4. State a necessary condition for a $9j$ symbol to be nonzero.

Solution

Each row and each column must satisfy angular momentum triangle rules. For example, in

$$\left\{ \begin{matrix} j_1 & j_2 & j_{12}\\ j_3 & j_4 & j_{34}\\ j_{13} & j_{24} & J \end{matrix} \right\},$$

the triples $(j_1,j_2,j_{12})$, $(j_3,j_4,j_{34})$, $(j_{13},j_{24},J)$ and the three column triples must all be allowed couplings.