Wigner Symbols

Wigner symbols package angular-momentum coupling data. This page fixes the convention and gives selection rules; use a validated generated source for large tables.

Convention

Clebsch–Gordan coefficients and Wigner 3-j symbols are related by

$$\langle j_1m_1,j_2m_2\vert JM\rangle =(-1)^{j_1-j_2+M}\sqrt{2J+1} \begin{pmatrix} j_1&j_2&J\\ m_1&m_2&-M \end{pmatrix}.$$

This page uses the Condon–Shortley phase convention.

Symbol Roles

Symbol	Role	Typical Use
3-j	recouples two angular momenta to one total	Clebsch–Gordan coefficients, tensor operators
6-j	changes the order of coupling three angular momenta	recoupling identities and spin networks
9-j	compares two binary coupling schemes for four angular momenta	multi-particle angular momentum

3-j Selection Rules

Condition	Consequence
$m_1+m_2+m_3=0$	otherwise the 3-j symbol vanishes
$\lvert j_1-j_2\rvert\le j_3\le j_1+j_2$	triangle condition
$j_1+j_2+j_3$ integer	half-integer parity consistency
$\lvert m_i\rvert\le j_i$	magnetic quantum numbers must be valid

Useful special case:

$$\begin{pmatrix} j&j&0\\ m&-m&0 \end{pmatrix} =\frac{(-1)^{j-m}}{\sqrt{2j+1}}.$$

Common Mistakes

• Missing the phase factor relating 3-j symbols to Clebsch–Gordan coefficients.
• Comparing 6-j or 9-j symbols without checking the ordering of entries.
• Treating selection rules as sufficient for a nonzero value; some allowed entries can still vanish.
• Copying a table generated with a different phase convention.

Canonical Links

• Wigner Symbols Quick Reference
• Recoupling and Wigner Symbols
• Clebsch–Gordan Coefficients
• Wigner–Eckart Theorem
• Wigner–Eckart Theorem Explained
• Angular Momentum Algebra

Sources and Verification

• 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.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.