Angular Momentum Algebra

Formula

Angular-momentum components satisfy

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

Equivalently,

$$[J_x,J_y] = i\hbar J_z,$$ $$[J_y,J_z] = i\hbar J_x,$$ $$[J_z,J_x] = i\hbar J_y.$$

Define the Casimir operator

$$J^2 = J_x^2+J_y^2+J_z^2.$$

It commutes with every component:

$$[J^2,J_i]=0.$$

The standard simultaneous eigenstates obey

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

The allowed labels are

$$j = 0,\frac12,1,\frac32,\ldots,$$ $$m = -j,-j+1,\ldots,j.$$

For fixed $j$, the multiplet contains $2j+1$ states.

At a Glance

Quantity	Formula
Component algebra	$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$
Casimir	$J^2=J_x^2+J_y^2+J_z^2$
Casimir commutator	$[J^2,J_i]=0$
Magnitude eigenvalue	$\hbar^2j(j+1)$
Chosen-component eigenvalue	$\hbar m$
Ladder definitions	$J_\pm=J_x\pm iJ_y$
Ladder commutator	$[J_z,J_\pm]=\pm\hbar J_\pm$
Ladder action	$J_\pm\lvert j,m\rangle=\hbar\sqrt{j(j+1)-m(m\pm1)}\,\lvert j,m\pm1\rangle$
Multiplet dimension	$2j+1$
Rotation	$U(\mathbf n,\theta)=\exp[-i\theta\,\mathbf n\cdot\mathbf J/\hbar]$

Meaning

The operators $J_i$ are generators of rotations. Their noncommutativity mirrors the noncommutativity of rotations about different axes. The same algebra is realized by:

• orbital angular momentum $\mathbf L$;
• intrinsic spin $\mathbf S$;
• total angular momentum $\mathbf J$;
• sums of independent angular momenta.

Sharing the algebra does not make these realizations physically identical. Orbital angular momentum acts on spatial wavefunctions, spin acts on internal degrees of freedom, and total angular momentum combines several contributions.

Casimir and Compatible Labels

Because

$$[J^2,J_z]=0,$$

states can be labeled simultaneously by $j$ and $m$. In contrast,

$$[J_x,J_z] = -i\hbar J_y$$

is generally nonzero, so $J_x$ and $J_z$ cannot generally be sharp together.

The physical magnitude associated with a $j$ multiplet is

$$\sqrt{ \langle J^2\rangle } = \hbar \sqrt{j(j+1)},$$

not $\hbar j$. The projection on the chosen axis is $\hbar m$.

The choice of $z$ is conventional. For any unit vector $\mathbf n$, the component

$$J_{\mathbf n} = \mathbf n\cdot\mathbf J$$

has eigenvalues $\hbar m$ within the same $j$ representation.

Ladder Operators

Define

$$J_+ = J_x+iJ_y,$$ $$J_- = J_x-iJ_y.$$

Then

$$[J_z,J_\pm] = \pm\hbar J_\pm,$$ $$[J^2,J_\pm]=0,$$ $$[J_+,J_-] = 2\hbar J_z.$$

With normalized basis states and the standard phase choice,

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

Equivalently,

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

At the endpoints,

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

The ladder operators change $m$ but preserve $j$.

Product Identities

The useful factorizations are

$$J_-J_+ = J^2-J_z^2-\hbar J_z,$$ $$J_+J_- = J^2-J_z^2+\hbar J_z.$$

They give the ladder-state norms:

$$\left\lVert J_+\lvert j,m\rangle \right\rVert^2 = \hbar^2 \left[ j(j+1)-m(m+1) \right],$$ $$\left\lVert J_-\lvert j,m\rangle \right\rVert^2 = \hbar^2 \left[ j(j+1)-m(m-1) \right].$$

Positivity and termination of these norms produce the finite range $-j\leq m\leq j$.

Component Matrix Elements

Recover the Cartesian components from

$$J_x = \frac12 \left( J_++J_- \right),$$ $$J_y = \frac{1}{2i} \left( J_+-J_- \right).$$

The nonzero matrix elements in a fixed-$j$ basis are

$$\langle j,m'\rvert J_z \lvert j,m\rangle = \hbar m \delta_{m'm},$$ $$\langle j,m'\rvert J_+ \lvert j,m\rangle = \hbar \sqrt{ (j-m)(j+m+1) } \delta_{m',m+1},$$ $$\langle j,m'\rvert J_- \lvert j,m\rangle = \hbar \sqrt{ (j+m)(j-m+1) } \delta_{m',m-1}.$$

These formulas construct the matrices for any finite $j$ representation.

Spin-One Example

Order the basis as

$$\left( \lvert1,1\rangle, \lvert1,0\rangle, \lvert1,-1\rangle \right).$$

Then

$$J_z = \hbar \begin{pmatrix} 1&0&0\\ 0&0&0\\ 0&0&-1 \end{pmatrix},$$ $$J_+ = \hbar\sqrt2 \begin{pmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{pmatrix},$$ $$J_- = \hbar\sqrt2 \begin{pmatrix} 0&0&0\\ 1&0&0\\ 0&1&0 \end{pmatrix}.$$

The matrices satisfy

$$J_-=(J_+)^\dagger.$$

Changing the basis order changes the displayed matrices but not the algebra.

Expectations in an Angular-Momentum Eigenstate

In $\lvert j,m\rangle$,

$$\langle J_x\rangle = \langle J_y\rangle = 0,$$ $$\langle J_z\rangle = \hbar m,$$ $$\langle J_x^2\rangle = \langle J_y^2\rangle = \frac{\hbar^2}{2} \left[ j(j+1)-m^2 \right].$$

Thus

$$(\Delta J_x)^2 = (\Delta J_y)^2 = \frac{\hbar^2}{2} \left[ j(j+1)-m^2 \right],$$

while

$$\Delta J_z=0.$$

The Robertson relation gives

$$\Delta J_x \Delta J_y \geq \frac{\hbar}{2} \lvert \langle J_z\rangle \rvert.$$

For the extremal states $m=\pm j$, this bound is saturated.

Rotation Operator

A rotation by angle $\theta$ about the unit vector $\mathbf n$ is represented by

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

For a rotation about $z$,

$$U_z(\theta) \lvert j,m\rangle = e^{-im\theta} \lvert j,m\rangle.$$

The sign shown is the active-state convention. Passive coordinate rotations may be written with the opposite sign; operators, states, and coordinates must be transformed consistently.

Dimensionless Generators

Some mathematical and quantum-information sources define

$$T_i = \frac{J_i}{\hbar}.$$

Then

$$[T_i,T_j] = i\epsilon_{ijk}T_k,$$ $$T^2\lvert j,m\rangle = j(j+1) \lvert j,m\rangle,$$ $$T_z\lvert j,m\rangle = m\lvert j,m\rangle.$$

The operators $T_i$ are dimensionless. Mixing their commutator with the dimensionful eigenvalue formulas introduces incorrect powers of $\hbar$.

Orbital Angular Momentum

For a spinless particle in three-dimensional Cartesian space,

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

In position representation,

$$\mathbf L = -i\hbar \mathbf r\times\nabla.$$

Orbital angular momentum has

$$\ell = 0,1,2,\ldots,$$

with

$$m_\ell = -\ell,-\ell+1,\ldots,\ell.$$

Only integer $\ell$ occurs for ordinary single-valued scalar wavefunctions on three-dimensional space. Monopoles, anyonic settings, or altered configuration spaces require separate treatments.

The position-space angular functions are spherical harmonics:

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

Spin and Pauli Conventions

Intrinsic spin obeys the same algebra. For spin one-half,

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

so

$$[S_i,S_j] = i\hbar \epsilon_{ijk}S_k,$$

whereas

$$[\sigma_i,\sigma_j] = 2i \epsilon_{ijk}\sigma_k.$$

The eigenvalues of $S_z$ are $\pm\hbar/2$, while those of $\sigma_z$ are $\pm1$. Pauli matrices are dimensionless; physical spin operators carry angular-momentum units.

Half-integer $j$ labels are representations of $SU(2)$ and projective representations of $SO(3)$. Single-valued ordinary representations of $SO(3)$ alone have integer $j$. This is why orbital scalar wavefunctions use integer $\ell$, while spinors can carry half-integer spin.

Total Angular Momentum

If two independent angular momenta commute across subsystems,

$$[J_{1i},J_{2j}]=0,$$

then

$$\mathbf J = \mathbf J_1+\mathbf J_2$$

satisfies the same angular-momentum algebra.

The allowed total labels are

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

The transformation between uncoupled and coupled bases is given by Clebsch–Gordan coefficients. Their phases require a stated convention.

Symmetry and Conservation

If a Hamiltonian is fully rotationally invariant,

$$[H,J_i]=0$$

for all three components. Then

$$[H,J^2]=0$$

as well, and the energy eigenspaces can be organized into angular-momentum multiplets.

Axial symmetry may preserve only one component, commonly

$$[H,J_z]=0.$$

It does not imply full rotational invariance or degeneracy across all $m$ values.

Symbols

Symbol	Meaning	Units or range
$\mathbf J$	Angular-momentum operator vector	angular momentum
$J^2$	Casimir operator	angular momentum squared
$j$	Total-angular-momentum quantum number	$0,1/2,1,\ldots$
$m$	Projection quantum number	$-j,-j+1,\ldots,j$
$J_\pm$	Raising and lowering operators	angular momentum
$\epsilon_{ijk}$	Levi-Civita symbol	dimensionless
$\mathbf L$	Orbital angular momentum	angular momentum
$\mathbf S$	Spin angular momentum	angular momentum
$\sigma_i$	Pauli matrix	dimensionless
$U(\mathbf n,\theta)$	Rotation operator	dimensionless unitary

The letter $j$ can denote either a quantum number or, in unrelated contexts, a current density. The bra–ket label and angular-momentum context distinguish the uses.

Units and Dimensions

Each $J_i$ has angular-momentum units:

$$[J_i]=[\hbar].$$

Consequently,

$$[J^2]=[\hbar]^2, \qquad [J_\pm]=[\hbar].$$

The labels $j$, $m$, Levi-Civita symbol, Pauli matrices, and rotation angle are dimensionless.

Assumptions

• The operators generate rotations and obey the standard dimensionful algebra.
• The representation is unitary and the angular-momentum operators are self-adjoint.
• States $\lvert j,m\rangle$ are normalized simultaneous eigenstates of $J^2$ and the chosen component.
• Ladder-state phases use the standard Condon–Shortley-compatible convention.
• Tensor-product angular momenta act on independent factors when the addition formulas are used.
• Orbital integer restrictions assume ordinary scalar wavefunctions and standard spatial topology.

Validity and Limitations

The algebra applies to orbital, spin, and total angular momentum, but specific realization-dependent statements do not automatically transfer:

• $\mathbf L=\mathbf R\times\mathbf P$ describes orbital, not intrinsic, angular momentum.
• Integer $\ell$ for ordinary orbital motion does not prohibit half-integer spin.
• Pauli matrices are normalized differently from physical spin operators.
• Full $m$ degeneracy requires rotational invariance of the Hamiltonian, not merely the algebra.
• Coupling coefficients and spherical harmonics depend on phase conventions.
• Effective pseudospin may obey the algebra without representing literal mechanical angular momentum.

Calculation Checks

• The number of states in a $j$ multiplet must be $2j+1$.
• $m$ changes in integer steps and never exceeds $\pm j$.
• $J_+$ must annihilate the top state and $J_-$ the bottom state.
• $J_-=(J_+)^\dagger$ in an orthonormal basis.
• The Casimir matrix must equal $\hbar^2j(j+1)I$ within one irreducible multiplet.
• Component matrices must satisfy the defining commutators.
• $j$ and $m$ are dimensionless; operator eigenvalues carry $\hbar$.
• Spin-one-half matrices must distinguish $S_i$ from $\sigma_i$.
• A rotation about $z$ should give phase $e^{-im\theta}$ under the displayed active convention.

Minimal Ladder Check

For $j=1$ and $m=0$,

$$J_+\lvert1,0\rangle = \hbar \sqrt{ 1(1+1)-0(0+1) } \lvert1,1\rangle.$$

Therefore

$$J_+\lvert1,0\rangle = \hbar\sqrt2 \lvert1,1\rangle.$$

Applying $J_+$ once more gives zero because the state has reached $m=j$.

Derivation and Canonical Home

Angular Momentum Algebra owns the rotation-generator interpretation and Casimir structure. Eigenvalues of J² and Jz derives the allowed labels, and Ladder Operators derives the normalized actions.

SU(2) owns the group-theoretic representation structure.

Worked Examples

• Orbital Angular Momentum
• Spin as Intrinsic Angular Momentum
• Total Angular Momentum
• Spherical Harmonics
• Angular-Momentum Problems

Common Mistakes

• Using $\hbar j$ instead of $\hbar\sqrt{j(j+1)}$ for the magnitude.
• Confusing the quantum numbers $j$ and $m$.
• Assuming $J_x$, $J_y$, and $J_z$ can all be sharp.
• Forgetting the factor of $\hbar$ in ladder actions.
• Reversing the signs in $J_\pm=J_x\pm iJ_y$.
• Applying a ladder operator outside the allowed $m$ range.
• Assuming $J_\pm$ changes $j$ rather than $m$.
• Confusing $\sigma_i$ with $S_i=\hbar\sigma_i/2$.
• Applying integer orbital restrictions to intrinsic spin.
• Mixing basis order or phase conventions when copying matrices or Clebsch–Gordan coefficients.
• Inferring full rotational symmetry from conservation of only $J_z$.

Related Formulas

• Ladder-Operator Action
• Spin-Half Matrices
• Addition of Angular Momentum
• Wigner–Eckart Theorem
• Pauli-Matrix Table
• Spin and Pauli-Matrix Conventions
• Commutator Table

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, chs. 3 and 4.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 12 and 14.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.