Clebsch–Gordan Quick Reference

This page is a quick reference for using Clebsch–Gordan coefficients without losing signs, labels, or basis conventions. It is meant for table lookup and calculation hygiene. The canonical explanation is Clebsch–Gordan Coefficients, the table-reading guide is Clebsch–Gordan Tables and Conventions, and the low-spin table is Clebsch–Gordan Coefficients.

Convention

The uncoupled basis is

$$\lvert j_1,m_1\rangle \lvert j_2,m_2\rangle,$$

and the coupled basis is

$$\lvert j_1,j_2;J,M\rangle.$$

The Clebsch–Gordan coefficient is the overlap

$$C^{JM}_{j_1m_1,j_2m_2} \equiv \langle j_1,m_1;j_2,m_2\vert J,M\rangle.$$

The coupled state expands as

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

This volume uses the Condon–Shortley phase convention unless a page explicitly states otherwise. With real coefficients in this convention, signs are fixed by the ladder-operator convention and the chosen highest-weight phases.

Allowed Labels

For two angular momenta $j_1$ and $j_2$,

$$J = \lvert j_1-j_2\rvert, \lvert j_1-j_2\rvert+1, \ldots, j_1+j_2.$$

For each allowed $J$,

$$M=-J,-J+1,\ldots,J.$$

The magnetic quantum number rule is

$$M=m_1+m_2.$$

If this rule fails, the coefficient is zero. The dimension check is

$$(2j_1+1)(2j_2+1) = \sum_{J=\lvert j_1-j_2\rvert}^{j_1+j_2} (2J+1).$$

Use this check to catch missing total-$J$ sectors before consulting a table.

Orthogonality and Inverse Expansion

For fixed $j_1$ and $j_2$, the Clebsch–Gordan coefficients form a unitary change of basis:

$$\sum_{m_1,m_2} \left( C^{JM}_{j_1m_1,j_2m_2} \right)^* C^{J'M'}_{j_1m_1,j_2m_2} = \delta_{JJ'}\delta_{MM'}.$$

Completeness in the uncoupled basis gives

$$\sum_{J,M} C^{JM}_{j_1m_1,j_2m_2} \left( C^{JM}_{j_1m'_1,j_2m'_2} \right)^* = \delta_{m_1m'_1}\delta_{m_2m'_2}.$$

The inverse expansion is

$$\lvert j_1,m_1\rangle \lvert j_2,m_2\rangle = \sum_{J,M} \langle J,M\vert j_1,m_1;j_2,m_2\rangle \lvert j_1,j_2;J,M\rangle.$$

For real coefficients, the inverse uses the same numerical coefficients after matching labels. In a complex convention, use complex conjugates.

Table-Reading Workflow

Before using a coefficient table, check these items in order:

1. Identify whether the table lists Clebsch–Gordan coefficients or Wigner $3j$ symbols.
2. Verify the phase convention, usually Condon–Shortley in physics tables.
3. Check the order of $j_1$ and $j_2$.
4. Apply the zero tests $M=m_1+m_2$ and $\lvert j_1-j_2\rvert\leq J\leq j_1+j_2$.
5. Check normalization by summing the squared coefficients for a fixed coupled state.

The table home is Clebsch–Gordan Coefficients. For a compact formula card, use Addition of Angular Momentum.

Quick Examples

For two spin-$1/2$ systems,

$$\frac12\otimes\frac12 = 1\oplus0.$$

In the standard convention,

$$\lvert 1,0\rangle = \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle \right),$$

while

$$\lvert 0,0\rangle = \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle \right).$$

The relative sign distinguishes the triplet state from the singlet state. The physical interpretation is developed in Singlet and Triplet States.

For $j_1=1$ and $j_2=1/2$,

$$1\otimes\frac12 = \frac32\oplus\frac12.$$

The $M=3/2$ state has only one product-basis component:

$$\left\lvert \frac32,\frac32 \right\rangle = \lvert1,1\rangle \left\lvert\frac12,\frac12\right\rangle.$$

Lowering this state and normalizing generates the $J=3/2$ multiplet; orthogonality then fixes the $J=1/2$ states up to the convention-fixed signs.

Relation to Wigner 3-j Symbols

Clebsch–Gordan coefficients and Wigner $3j$ symbols contain the same coupling information. With the convention used here,

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

The $3j$ symbol is often better for identities and symmetry relations. The Clebsch–Gordan coefficient is often better for changing basis between uncoupled and coupled states. Do not use a $3j$ table as a Clebsch–Gordan table without the phase and normalization factor.

Coupling Order Warning

For two angular momenta, the main ordering issue is whether the table uses $j_1,j_2$ or $j_2,j_1$. For three or more angular momenta, one must also specify which pair is coupled first.

For example,

$$\left| (j_1j_2)j_{12},j_3;JM \right\rangle$$

and

$$\left| j_1,(j_2j_3)j_{23};JM \right\rangle$$

are different coupled bases. Their change of basis involves Wigner $6j$ symbols, not merely a Clebsch–Gordan table. Use Recoupling and Wigner Symbols when the problem has more than two angular momenta.

Which Page Should I Open?

Task	Best starting page
Understand coupled versus uncoupled bases	Coupled and Uncoupled Bases
Learn what the coefficients mean	Clebsch–Gordan Coefficients
Resolve table and phase conventions	Clebsch–Gordan Tables and Conventions
Look up low-spin coefficients	Clebsch–Gordan Coefficients Table
Check allowed $J$ values and $M$ rules	Addition of Angular Momentum
Interpret the singlet and triplet	Singlet and Triplet States
Convert to Wigner $3j$ notation	Wigner Symbols
Change coupling order for three angular momenta	Recoupling and Wigner Symbols

Common Mistakes

• Forgetting that Clebsch–Gordan coefficients are basis-change amplitudes, not energy-level data.
• Using a coefficient even though $M\ne m_1+m_2$.
• Comparing signs from tables with different phase conventions.
• Swapping $j_1$ and $j_2$ without checking the table convention.
• Treating $3j$ symbols and Clebsch–Gordan coefficients as identical.
• Omitting multiplicity or intermediate-coupling labels in problems with more than two angular momenta.

References

• 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.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.

Quick Checks

1. Which total angular momenta appear in $2\otimes1$?

Solution

The allowed total angular momenta run from

$$\lvert2-1\rvert=1$$

to

$$2+1=3$$

in integer steps. Therefore

$$2\otimes1=3\oplus2\oplus1.$$

2. Can $\langle1,1;1,-1\vert1,1\rangle$ be nonzero?

Solution

The magnetic quantum number rule requires

$$M=m_1+m_2.$$

Here $m_1=1$ and $m_2=-1$, so $m_1+m_2=0$, while the coupled state has $M=1$. The coefficient is therefore zero.

3. Why is a $3j$ table not automatically a Clebsch–Gordan table?

Solution

The two notations differ by a phase and a normalization factor:

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

Using a $3j$ value directly as a Clebsch–Gordan coefficient generally gives the wrong sign or magnitude.