Angular Momentum Operator

The angular momentum operator is written $\mathbf J$ for a generic or total angular momentum. Orbital angular momentum is usually $\mathbf L$ , spin angular momentum is $\mathbf S$ , and components are written $J_x,J_y,J_z$ or $J_i$ .

Default Meaning

The components of any angular momentum operator satisfy

[J_i,J_j] = i\hbar\sum_k\epsilon_{ijk}J_k.

The squared magnitude is

J^2=J_x^2+J_y^2+J_z^2,

and the standard simultaneous eigenstates obey

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.

Mathematical Type

Angular momentum components are self-adjoint observables on their appropriate domains. They are also generators of rotations through

U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{\hbar}\theta\,\hat{\mathbf n}\cdot\mathbf J \right).

Units

The operators $J_i$ , $L_i$ , and $S_i$ have units of angular momentum. The labels $j$ , $\ell$ , $s$ , and $m$ are dimensionless; the eigenvalues carry the powers of $\hbar$ .

Common Variants

$\mathbf L$ : orbital angular momentum, often $\mathbf R\times\mathbf P$ .
$\mathbf S$ : spin angular momentum acting on internal spin degrees of freedom.
$\mathbf J$ : total angular momentum or a generic angular momentum operator.
$J_\pm=J_x\pm iJ_y$ : raising and lowering operators.

Convention Warnings

Do not confuse the operator $\mathbf J$ with the quantum number $j$ .
$L$ can also mean length, Lagrangian, or a loss operator in other contexts.
$S$ can also mean action, entropy, or an $S$ matrix.
Only one component, conventionally $J_z$ , is diagonalized with $J^2$ in the standard basis.

Canonical Links

References

J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
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.