Clebsch–Gordan Tables and Conventions

Clebsch–Gordan tables are useful only after the notation and phase convention are fixed. The same physical change of basis can be printed with different signs, different orderings of $j_1,j_2$, or Wigner $3j$ symbols instead of Clebsch–Gordan coefficients. This page is a table-reading and convention guide.

The conceptual definition is in Clebsch–Gordan Coefficients. The low-spin reference table is Clebsch–Gordan Coefficients. This page explains how to move between them without losing signs.

Canonical Split

Need	Canonical home
Meaning of the coefficients	Clebsch–Gordan Coefficients
Table lookup workflow	this page and Clebsch–Gordan Quick Reference
Low-spin coefficient table	Reference table
Wigner $3j$, $6j$, and $9j$ notation	Recoupling and Wigner Symbols and Wigner Symbols

Large coefficient atlases and symbolic generation belong outside this page. The purpose here is to make conventions explicit.

Notation Used Here

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

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

Some tables suppress commas and write $\lvert j_1m_1\rangle\lvert j_2m_2\rangle$ or $\lvert JM\rangle$. That shorthand is harmless only when $j_1$ and $j_2$ have already been fixed.

Phase Convention

This volume uses the Condon–Shortley phase convention unless a page explicitly states otherwise. In this convention the ladder operators act as

$$J_\pm\lvert j,m\rangle = \hbar \sqrt{j(j+1)-m(m\pm1)} \, \lvert j,m\pm1\rangle,$$

with the square-root coefficient chosen positive. Coupled highest-weight states are then fixed by the standard positive leading coefficient choice.

The numerical signs in a table are not convention-free facts. If a source rephases a basis state,

$$\lvert j,m\rangle \mapsto e^{i\alpha_{jm}} \lvert j,m\rangle,$$

then coefficients involving that state change by compensating phases. Probabilities and complete state expansions remain physically equivalent, but mixing tables from different conventions in one calculation can produce wrong signs.

Zero Tests Before Table Lookup

Before consulting a table, apply the selection rules.

The magnetic quantum numbers must add:

$$M=m_1+m_2.$$

The total angular momentum must satisfy the triangle rule:

$$|j_1-j_2|\leq J\leq j_1+j_2.$$

The labels must also satisfy

$$-j_a\leq m_a\leq j_a, \qquad -J\leq M\leq J.$$

If any of these fail, the coefficient is zero. Do not spend time searching a table for forbidden entries.

Reading a Coupled-State Expansion

A table may give the expansion of a coupled state directly. For example, for $j_1=1$ and $j_2=1/2$, a Condon–Shortley table may list

$$\left\lvert \frac32,\frac12 \right\rangle = \sqrt{\frac23} \lvert1,0\rangle \left\lvert\frac12,\frac12\right\rangle + \sqrt{\frac13} \lvert1,1\rangle \left\lvert\frac12,-\frac12\right\rangle.$$

This means, among other things,

$$\left\langle 1,0;\frac12,\frac12 \middle| \frac32,\frac12 \right\rangle = \sqrt{\frac23}.$$

The same line also means

$$\left\langle 1,1;\frac12,-\frac12 \middle| \frac32,\frac12 \right\rangle = \sqrt{\frac13}.$$

It does not give coefficients for a different ordering of the two angular momenta unless the table says so.

Order of j1 and j2

Interchanging the two angular momenta changes the coefficient by a convention-fixed phase:

$$\langle j_2,m_2;j_1,m_1\vert J,M\rangle = (-1)^{j_1+j_2-J} \langle j_1,m_1;j_2,m_2\vert J,M\rangle.$$

The exponent $j_1+j_2-J$ is always an integer for allowed angular momentum coupling. This symmetry is useful, but it is not permission to ignore ordering. Many sign mistakes come from reading a table printed for $j_1,j_2$ as if it were printed for $j_2,j_1$.

For two identical spin-$1/2$ factors, this rule explains why the triplet states are symmetric under interchange while the singlet is antisymmetric.

Relation to Wigner 3j Symbols

Some references tabulate Wigner $3j$ symbols instead of Clebsch–Gordan coefficients. With the convention used here,

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

Equivalently,

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

The $3j$ notation is often cleaner for symmetry identities because it treats the three angular momenta more symmetrically. Clebsch–Gordan notation is often cleaner when the task is to change basis between uncoupled and coupled states. They are related, not identical.

Table-Reading Checklist

Use this checklist before copying a number:

1. Confirm whether the source is listing Clebsch–Gordan coefficients, $3j$ symbols, or coupled-state expansions.
2. Check that the source uses the Condon–Shortley phase convention, or translate its convention explicitly.
3. Verify the order of $j_1$ and $j_2$.
4. Apply $M=m_1+m_2$ and the triangle rule.
5. Check whether the table is normalized as coupled states expanded in product states or the inverse expansion.
6. Keep the same convention throughout a calculation.

A correct coefficient with the wrong convention is still the wrong number for your calculation.

When a Table Is Not Enough

Tables are ideal for low angular momenta and hand calculations. They become awkward when:

• several angular momenta are coupled in different orders;
• large $j$ values make printed tables unwieldy;
• a symbolic expression is needed for arbitrary labels;
• a numerical calculation must generate many coefficients consistently.

For three or more angular momenta, specify intermediate labels such as $j_{12}$ before using coefficients. Changing coupling order is a recoupling problem and usually involves Wigner $6j$ or $9j$ symbols; see Recoupling and Wigner Symbols.

Common Mistakes

• Copying a $3j$ symbol as if it were a Clebsch–Gordan coefficient.
• Swapping $j_1$ and $j_2$ without the phase factor.
• Comparing signs from two sources before checking their phase conventions.
• Forgetting that tables often omit zero entries.
• Reading a coupled-state expansion backward without complex conjugation in a complex convention.
• Using a two-angular-momentum table for a problem that needs an intermediate coupling label.
• Treating signs as directly observable when only the complete convention-fixed state has physical meaning.

Cross-Links

• Clebsch–Gordan Coefficients
• Clebsch–Gordan Quick Reference
• Clebsch–Gordan Coefficient Table
• Coupled and Uncoupled Bases
• Total Angular Momentum
• Two Spin-1/2 Particles
• Singlet and Triplet States
• Recoupling and Wigner Symbols
• Wigner Symbols
• Addition of Angular Momentum Formula Card
• Toolkit Clebsch–Gordan Coefficients

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

Exercises

1. Decide whether $\langle 1,1;\tfrac12,\tfrac12\vert\tfrac12,\tfrac12\rangle$ can be nonzero.

Solution

The magnetic quantum number rule requires

$$M=m_1+m_2.$$

Here

$$m_1+m_2=1+\frac12=\frac32,$$

but the coupled state has $M=1/2$. The coefficient is therefore zero.

2. Convert a Clebsch–Gordan coefficient to a $3j$ symbol. If

$$\langle j_1,m_1;j_2,m_2\vert J,M\rangle=A,$$

what is

$$\begin{pmatrix} j_1&j_2&J\\ m_1&m_2&-M \end{pmatrix}$$

in the convention used on this page?

Solution

Use

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

Therefore

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

3. For $j_1=1$, $j_2=1/2$, and $J=1/2$, what phase relates the coefficient with $j_1,j_2$ to the coefficient with $j_2,j_1$?

Solution

The interchange formula gives the phase

$$(-1)^{j_1+j_2-J} = (-1)^{1+\frac12-\frac12} = (-1)^1 = -1.$$

Thus

$$\langle \tfrac12,m_2;1,m_1\vert\tfrac12,M\rangle = - \langle 1,m_1;\tfrac12,m_2\vert\tfrac12,M\rangle.$$

4. Why can two correct tables disagree by signs?

Solution

Clebsch–Gordan coefficients depend on the phase convention for the basis states. If a source rephases one or more angular momentum basis states, the expansion coefficients change by compensating phases. The physical state and transition probabilities are unchanged when the convention is used consistently, but copying signs from incompatible conventions into one calculation is not valid.