Clebsch–Gordan Coefficients

Canonical treatment: Clebsch–Gordan Coefficients owns the construction algorithm, coupled-state examples, applications, exercises, and references. This Toolkit page retains the representation-theory prerequisites and convention ledger.

For irreducible $SU(2)$ representations,

$$V_{j_1}\otimes V_{j_2} \cong \bigoplus_{J=\lvert j_1-j_2\rvert}^{j_1+j_2}V_J.$$

The Clebsch–Gordan coefficients are the entries of the unitary basis transformation

$$\lvert j_1,j_2;J,M\rangle = \sum_{m_1,m_2} C^{JM}_{j_1m_1,j_2m_2} \lvert j_1,m_1\rangle\lvert j_2,m_2\rangle.$$

Structural Identities

The weight selection rule is

$$C^{JM}_{j_1m_1,j_2m_2}=0 \quad\text{unless}\quad M=m_1+m_2.$$

Unitarity gives orthonormality and completeness:

$$\sum_{m_1,m_2} C^{JM*}_{j_1m_1,j_2m_2} C^{J'M'}_{j_1m_1,j_2m_2} = \delta_{JJ'}\delta_{MM'},$$ $$\sum_{J,M} C^{JM}_{j_1m_1,j_2m_2} C^{JM*}_{j_1m'_1,j_2m'_2} = \delta_{m_1m'_1}\delta_{m_2m'_2}.$$

These identities are convention independent. Individual coefficient signs are not.

Convention Ledger

The commonly used Condon–Shortley convention chooses real coefficients and fixes the highest-weight phase. Other internally consistent phase choices describe the same basis change. Never combine coefficients or $3j$ symbols from different tables without checking conventions.

In a standard convention,

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

Routing

• Tensor-Product Representations explains why reducible products must be decomposed.
• Two Spin-1/2 Particles gives the smallest physical example.
• Wigner Symbols routes to $3j$, $6j$, and recoupling notation.
• Selection Rules and Transition Rates uses the coefficients in matrix-element organization.