Angular Momentum Identity Index

Angular-momentum calculations use a small set of commutator, ladder, and coupling identities. This page gives their conventions and links each formula to a derivation or table.

The one-page table hub is Angular Momentum Tables. The conceptual home is Angular Momentum Algebra. The formulas below use normalized states and sum repeated Cartesian indices.

Core Algebra

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

Need	Identity	Derivation or table
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

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

For orbital applications, see:

• 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 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:

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

The allowed angular labels also require $j+j'+k\in\mathbb Z$ and $|q|\le k$. A permitted angular coefficient still need not have a nonzero reduced matrix element. The immediate magnetic and triangle 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.

References

• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.

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 $q=0$,

$$m'=m.$$