Angular Momentum Identity Index

This index points to the canonical angular-momentum identities used throughout the volume. Use it when you know roughly what identity you need, but need the right formula card, table, or derivation page.

The one-page table hub is Angular Momentum Tables. The conceptual home is Angular Momentum Algebra. This page sits between them: it records the main identity families and tells you where each one belongs.

Core Algebra

Use these identities when the calculation is about the $SU(2)$ angular-momentum algebra itself.

Need	Identity	Canonical target
Commutators	$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$	Angular Momentum Algebra
Casimir eigenvalue	$J^2\lvert j,m\rangle=\hbar^2j(j+1)\lvert j,m\rangle$	Eigenvalues of $J^2$ and $J_z$
Component eigenvalue	$J_z\lvert j,m\rangle=\hbar m\lvert j,m\rangle$	Angular Momentum Algebra
Multiplet range	$m=-j,-j+1,\ldots,j$	Angular Momentum Tables
Dimension	$\dim\mathcal H_j=2j+1$	SU(2)

The convention is $\epsilon_{xyz}=+1$ and

$$J_\pm=J_x\pm iJ_y.$$

If a sign differs from a table, first check the ladder-operator convention and the Condon–Shortley phase convention.

Ladder Operators

Use ladder identities when you are moving within one fixed multiplet. The ladder operators obey

$$[J_z,J_\pm]=\pm\hbar J_\pm, \qquad [J_+,J_-]=2\hbar J_z.$$

Their normalized action is

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

Endpoint states satisfy

$$J_+\lvert j,j\rangle=0, \qquad J_-\lvert j,-j\rangle=0.$$

Use Ladder-Operator Action for the formula card and Ladder Operators for the derivation and interpretation.

Orbital Angular Momentum

Use orbital identities when the operators act on wavefunctions on ordinary space:

$$\mathbf L = \mathbf R\times\mathbf P.$$

The spherical harmonics satisfy

$$L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m, \qquad L_zY_\ell^m = \hbar mY_\ell^m.$$

The canonical explanatory path is:

• Orbital Angular Momentum
• Spherical Harmonics
• Central Potentials
• Hydrogen Atom Angular Structure

For lookup tables, use Spherical Harmonics.

Spin Matrices

Use spin matrix tables when you need explicit finite-dimensional matrices rather than abstract algebra.

For spin-$1/2$,

$$S_i = \frac{\hbar}{2}\sigma_i,$$

and

$$\sigma_i\sigma_j = \delta_{ij}I + i\epsilon_{ijk}\sigma_k.$$

For spin-$j$ matrices, use the basis ordered by $m=j,j-1,\ldots,-j$ unless a table states otherwise. The table home is Spin Matrices. The Pauli matrix home is Pauli Matrices, with conceptual discussion in Pauli Matrices.

Addition of Angular Momentum

For two angular momenta $j_1$ and $j_2$, the allowed total angular momenta are

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

For each $J$,

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

and Clebsch-Gordan coefficients vanish unless

$$M=m_1+m_2.$$

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 Addition of Angular Momentum for the compact formula card, Clebsch-Gordan Coefficients for the canonical explanation, and Clebsch-Gordan Coefficients for table lookup.

Coupled and Uncoupled Bases

The coupled basis expands as

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

Use:

• Coupled and Uncoupled Bases for the basis change;
• Two Spin-Half Particles for the singlet-triplet example;
• Singlet and Triplet States for the physical interpretation.

Do not infer energy ordering from the allowed $J$ values alone. Energies require a Hamiltonian, such as a spin-spin or spin–orbit coupling.

Wigner Symbols and Recoupling

Use Wigner symbols when angular momenta are coupled in different orders or when tensor-operator matrix elements are being organized.

The Clebsch-Gordan and $3j$ conventions are related by

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

The $3j$ symbol vanishes unless

$$m_1+m_2+m_3=0$$

and the triangle conditions are satisfied. Use Wigner Symbols for lookup conventions and Recoupling and Wigner Symbols for the conceptual role of $6j$ and $9j$ symbols.

Tensor Operators and Selection Rules

For an irreducible spherical tensor $T_q^{(k)}$, the Wigner–Eckart theorem separates angular dependence from dynamics:

$$\begin{aligned} &\langle\alpha',j',m'| T_q^{(k)} |\alpha,j,m\rangle \\ &\quad = (-1)^{j'-m'} \begin{pmatrix} j'&k&j\\ -m'&q&m \end{pmatrix} \langle\alpha',j'\lVert T^{(k)}\rVert\alpha,j\rangle. \end{aligned}$$

The immediate angular selection rules are

$$m'=m+q, \qquad \lvert j-k\rvert\le j'\le j+k.$$

Use Wigner–Eckart Theorem for the theorem and Selection Rules for the broader symmetry logic.

Which Page Should I Open?

Task	Best starting page
Check a sign in $[J_i,J_j]$	Angular Momentum Algebra Formula Card
Derive ladder coefficients	Ladder Operators
Look up explicit spin matrices	Spin Matrices
Look up $Y_{\ell m}$	Spherical Harmonics Table
Couple two angular momenta	Clebsch-Gordan Coefficients
Practice solved angular-momentum calculations	Angular Momentum Problems
Convert CG coefficients to $3j$ symbols	Wigner Symbols
Apply tensor-operator selection rules	Wigner–Eckart Theorem
Work a spin–orbit problem	Spin–Orbit Coupling

Common Mistakes

• Mixing $\ell$, $s$, $j$, and $J$ without stating which angular momentum is being used.
• Treating $m$ as having units; $\hbar m$ is the $J_z$ eigenvalue.
• Forgetting that ladder operators change $m$, not $j$.
• Comparing Clebsch-Gordan tables without checking phase convention.
• Using orbital selection rules after spin–orbit coupling has changed the good quantum numbers.
• Treating a nonzero angular coefficient as proof that a physical matrix element is large.
• Forgetting that parity, exchange symmetry, and other symmetries can add selection rules beyond angular momentum.

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.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.

Quick Checks

1. What does $J_+$ do to $\lvert j,j\rangle$?

Solution

It annihilates the highest-weight state:

$$J_+\lvert j,j\rangle=0.$$

The square-root coefficient contains

$$j(j+1)-j(j+1)=0.$$

2. Which total angular momenta appear in $1\otimes\frac12$?

Solution

The allowed values run from

$$\left\lvert1-\frac12\right\rvert=\frac12$$

to

$$1+\frac12=\frac32$$

in integer steps. Therefore

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

3. A rank-$1$ tensor component has $q=0$. What magnetic selection rule follows?

Solution

The Wigner–Eckart magnetic rule is

$$m'=m+q.$$

For $q=0$,

$$m'=m.$$