Angular Momentum Tables

This page is a compact lookup hub for angular momentum. It fixes the convention, records the identities most often needed in calculations, and points to the detailed tables for spin matrices, spherical harmonics, Clebsch–Gordan coefficients, and Wigner symbols.

For derivations and physical interpretation, use the canonical pages linked below. This page is for checking signs, labels, and table locations.

Scope

Use this table when you need to identify:

• the meaning of $j$, $\ell$, $s$, $m$, $J$, $L$, and $S$;
• the standard eigenvalue equations for angular momentum;
• the ladder-operator convention;
• the allowed values in angular-momentum addition;
• which table contains the coefficient or matrix you need.

This page does not replace the canonical treatment of angular momentum algebra, spin, addition of angular momentum, or spin–orbit coupling.

Convention

Angular momentum components satisfy

$$[J_i,J_j] =i\hbar\sum_k\epsilon_{ijk}J_k, \qquad \epsilon_{xyz}=+1.$$

The standard basis diagonalizes $J^2$ and $J_z$:

$$J^2\lvert j,m\rangle =\hbar^2j(j+1)\lvert j,m\rangle, \qquad J_z\lvert j,m\rangle =\hbar m\lvert j,m\rangle.$$

The labels $j$ and $m$ are dimensionless. The eigenvalues carry the factors of $\hbar$.

The ladder operators are

$$J_\pm=J_x\pm iJ_y,$$

with action

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

Spherical harmonics and coupling coefficients use the Condon–Shortley phase convention unless a page declares otherwise.

Labels and Operators

Symbol	Meaning	Typical eigenstate label	Notes
$J$	generic or total angular momentum	$\lvert j,m\rangle$	Use when the source of angular momentum is not being distinguished
$L$	orbital angular momentum	$\lvert \ell,m\rangle$ or $Y_{\ell m}$	Integer $\ell$ for ordinary orbital wavefunctions
$S$	spin angular momentum	$\lvert s,m_s\rangle$	$s$ may be integer or half-integer
$J=L+S$	total angular momentum from orbital plus spin	$\lvert j,m\rangle$	Common in atoms and spin–orbit coupling
$J=J_1+J_2$	total angular momentum of two subsystems	$\lvert j_1,j_2;J,M\rangle$	Used in Clebsch–Gordan expansions

For a fixed $j$,

$$m=-j,-j+1,\ldots,j, \qquad \dim\mathcal H_j=2j+1.$$

Core Identities

Quantity	Formula	Use
Casimir eigenvalue	$J^2\lvert j,m\rangle=\hbar^2j(j+1)\lvert j,m\rangle$	Identifies the angular-momentum multiplet
Component eigenvalue	$J_z\lvert j,m\rangle=\hbar m\lvert j,m\rangle$	Labels the chosen quantization axis
Ladder action	$J_\pm\lvert j,m\rangle=\hbar\sqrt{j(j+1)-m(m\pm1)}\lvert j,m\pm1\rangle$	Builds a multiplet from highest or lowest weight
Raising/lowering inverse	$J_x=(J_++J_-)/2$	Converts ladder operators to Cartesian components
Raising/lowering inverse	$J_y=(J_+-J_-)/(2i)$	Fixes the sign convention for $J_y$
Orbital eigenfunctions	$L^2Y_{\ell m}=\hbar^2\ell(\ell+1)Y_{\ell m}$	Central potentials and rotors
Orbital component	$L_zY_{\ell m}=\hbar mY_{\ell m}$	Azimuthal quantum number

Addition Rules

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 allowed $J$,

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

The magnetic quantum numbers obey the selection rule

$$M=m_1+m_2.$$

The dimension-counting check is

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

Which Table To Use

Need	Open this table	Canonical explanation
Pauli matrices and spin-$1/2$ operators	Pauli Matrices	Pauli Matrices
Spin-$j$ matrices and ladder matrices	Spin Matrices	Ladder Operators
Low-order $Y_{\ell m}(\theta,\phi)$	Spherical Harmonics	Spherical Harmonics
Coupled two-spin states	Clebsch–Gordan Coefficients	Clebsch–Gordan Coefficients
3-j, 6-j, and 9-j conventions	Wigner Symbols	Wigner-Eckart Theorem
Compact formula card	Angular Momentum Algebra	Angular Momentum Algebra

Common Mistakes

• Treating $j$ and $m$ as angular momenta with units. They are dimensionless labels; the eigenvalues carry $\hbar$.
• Reversing the sign convention in $J_\pm=J_x\pm iJ_y$.
• Comparing Clebsch–Gordan coefficients from tables with different phase conventions.
• Confusing orbital $\ell$ with total $j$ in spin–orbit problems.
• Assuming the allowed total $J$ values are energy eigenvalues before a Hamiltonian has been specified.
• Forgetting that $L_x$, $L_y$, and $L_z$ cannot be simultaneously diagonalized.

Canonical Links

• Angular Momentum Identity Index
• Angular Momentum Conventions
• Angular Momentum Algebra
• Ladder Operators
• Spherical Harmonics
• Addition of Angular Momentum
• Spin–Orbit Coupling
• SU(2)

References

• 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.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.

Quick Checks

1. What are the allowed $m$ values for $j=2$?

Solution

For fixed $j=2$,

$$m=-2,-1,0,1,2.$$

There are $2j+1=5$ states in the multiplet.

2. List the allowed total angular momenta for $j_1=3/2$ and $j_2=1$.

Solution

The allowed values run from

$$\lvert j_1-j_2\rvert=\frac12$$

to

$$j_1+j_2=\frac52$$

in steps of one. Therefore

$$J=\frac12,\frac32,\frac52.$$

The dimension check is

$$(2\cdot3/2+1)(2\cdot1+1) =4\cdot3=12,$$

and

$$(2\cdot1/2+1) +(2\cdot3/2+1) +(2\cdot5/2+1) =2+4+6=12.$$