# Angular momentum operator

In quantum mechanics, an **angular [momentum operator](https://www.edgepedia.org/momentum-operator)** is a vector operator \( \hat{\mathbf{J}} \) whose components generate spatial rotations of a quantum state and whose components obey the commutation relations \( [\hat{J}_i, \hat{J}_j] = i\hbar \sum_k \varepsilon_{ijk} \hat{J}_k \), where \( \varepsilon_{ijk} \) is the [Levi-Civita symbol](https://www.edgepedia.org/levi-civita-symbol)<sup>[1](https://buks.net.technion.ac.il/files/2024/10/qm_lecture_notes.pdf)</sup><sup> • </sup><sup>[2](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture7.html)</sup>. Three versions appear in practice: the orbital angular momentum \( \hat{\mathbf{L}} \) built from position and momentum, the intrinsic spin \( \hat{\mathbf{S}} \), and their sum, the total angular momentum \( \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} \)<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup>.

| Key fact | Detail |
|---|---|
| Defining algebra | \( [\hat{J}_i, \hat{J}_j] = i\hbar \varepsilon_{ijk} \hat{J}_k \); the same relations hold for \( \hat{\mathbf{L}} \), \( \hat{\mathbf{S}} \), and \( \hat{\mathbf{J}} \)<sup>[1](https://buks.net.technion.ac.il/files/2024/10/qm_lecture_notes.pdf)</sup><sup> • </sup><sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup> |
| Role in rotations | A rotation by angle \( \alpha \) about a unit axis \( \mathbf{e}_n \) is \( \hat{R}(\alpha, \mathbf{e}_n) = \exp[-i\alpha\, \mathbf{e}_n \cdot \hat{\mathbf{L}}/\hbar] \)<sup>[2](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture7.html)</sup> |
| Spectrum | \( \hat{J}^2 \) has eigenvalues \( j(j+1)\hbar^2 \) and \( \hat{J}_z \) has eigenvalues \( m\hbar \), with \( m = -j, -j+1, \ldots, j \); \( j \) is an integer or half-integer<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup> |
| Degeneracy | For a given \( j \), the eigenvalue \( j(j+1)\hbar^2 \) is \( (2j+1) \)-fold degenerate<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup> |
| Orbital form | \( \hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x \), which in spherical coordinates is \( -i\hbar\, \partial/\partial\phi \)<sup>[5](https://www.feynmanlectures.caltech.edu/III_20.html)</sup><sup> • </sup><sup>[6](https://math.hodapp.com/lectures/physics/sicm-quantum-angular-momentum.html)</sup> |
| Spin-1/2 matrices | \( \hat{S}_i = \frac{\hbar}{2}\sigma_i \) with the Pauli matrices \( \sigma_i \), acting on a two-dimensional spinor space<sup>[7](https://www.physik.uni-leipzig.de/~rudolph/qm/qmgr.pdf)</sup> |
| Addition rule | For \( \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} \), the total quantum number satisfies \( |l-s| \leq j \leq l+s \)<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup> |

## Generator of rotations

The link between angular momentum and rotation is the operator's defining property. Acting on a state with no internal spin variables, a rotation by an infinitesimal angle \( \epsilon \) about the \( z \)-axis is written \( \hat{R}_z(\epsilon)|\psi\rangle = (1 + \frac{i}{\hbar}\epsilon \hat{L}_z)|\psi\rangle \), and a finite rotation follows by exponentiation, \( \hat{R}(\alpha, \mathbf{e}_n) = \exp[-i\alpha\, \mathbf{e}_n \cdot \hat{\mathbf{L}}/\hbar] \)<sup>[5](https://www.feynmanlectures.caltech.edu/III_20.html)</sup><sup> • </sup><sup>[2](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture7.html)</sup>. Under this transformation a vector operator \( \hat{\mathbf{v}} \) (position, momentum, or \( \hat{\mathbf{L}} \) itself) rotates as an ordinary three-vector, which is encoded in the commutator \( [\hat{v}_i, \hat{L}_j] = i\hbar \sum_k \varepsilon_{ijk} \hat{v}_k \)<sup>[2](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture7.html)</sup>.

Because rotations about different axes do not commute, the generators cannot commute either, and the commutator \( [\hat{L}_i, \hat{L}_j] = i\hbar \varepsilon_{ijk} \hat{L}_k \) expresses exactly this failure<sup>[1](https://buks.net.technion.ac.il/files/2024/10/qm_lecture_notes.pdf)</sup>. The non-commutativity is not a nuisance but the source of the quantization: it is what forces the discrete spectrum described below.

## Orbital angular momentum

For a particle moving in space, the orbital operator is built from position and momentum: \( \hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x \), with cyclic permutations for the other components<sup>[5](https://www.feynmanlectures.caltech.edu/III_20.html)</sup>. In the coordinate representation, where \( \hat{p}_i = \frac{\hbar}{i} \partial_i \), the components become differential operators; in spherical coordinates \( \hat{L}_z = -i\hbar\, \partial/\partial\phi \), and the \( x \) and \( y \) components mix \( \theta \) and \( \phi \) derivatives<sup>[6](https://math.hodapp.com/lectures/physics/sicm-quantum-angular-momentum.html)</sup><sup> • </sup><sup>[8](https://www.ippp.dur.ac.uk/~krauss/Lectures/TheoreticalPhysics_2B/QM/Lecture15.pdf)</sup>. The eigenfunctions of \( \hat{L}^2 \) and \( \hat{L}_z \) are the spherical harmonics, and for orbital angular momentum the quantum number \( l \) takes only integer values \( 0, 1, 2, \ldots \)<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>.

## Spin

Experiment shows that particles such as the electron carry an internal angular momentum, spin, that is not the orbital motion of internal parts; treating it that way leads to contradictions<sup>[9](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.01%3A_Angular_Momentum_Operator_Algebra)</sup>. Spin operators obey the same commutation relations as \( \hat{\mathbf{L}} \)<sup>[10](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/09%3A_Spin_Angular_Momentum/9.01%3A_Spin_Operators)</sup>, but since spin is not defined in terms of position and momentum, its eigenstates are not functions on space; they are vectors in an internal space, and the operators are represented by matrices<sup>[11](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_%28Cappellaro%29/04%3A_Energy_Levels/4.02%3A_Quantum_Mechanics_in_3D_-_Angular_momentum)</sup>.

For a spin-\( \frac{1}{2} \) particle the spinor space is two-dimensional, spanned by the spin-up and spin-down eigenstates of \( \hat{S}_3 \), and the operators take the form \( \hat{S}_i = \frac{\hbar}{2}\sigma_i \)<sup>[7](https://www.physik.uni-leipzig.de/~rudolph/qm/qmgr.pdf)</sup>. Each elementary particle species has a fixed, immutable spin quantum number \( s \): pi mesons have spin 0, electrons spin \( \frac{1}{2} \), photons spin 1, and gravitons spin 2<sup>[11](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_%28Cappellaro%29/04%3A_Energy_Levels/4.02%3A_Quantum_Mechanics_in_3D_-_Angular_momentum)</sup>.

## Spectrum from the commutation relations

Much of the physics follows from the algebra alone. Since \( \hat{J}^2 \) commutes with every component of \( \hat{\mathbf{J}} \), simultaneous eigenkets of \( \hat{J}^2 \) and one component, conventionally \( \hat{J}_z \), can be constructed; no basis can simultaneously diagonalize more than one component<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>. The ladder operators \( \hat{J}_\pm = \hat{J}_x \pm i\hat{J}_y \) satisfy \( [\hat{J}_z, \hat{J}_\pm] = \pm\hbar\, \hat{J}_\pm \) and \( [\hat{J}^2, \hat{J}_\pm] = 0 \), so \( \hat{J}_+ \) raises \( m \) by one unit without changing \( j \), and \( \hat{J}_- \) lowers it<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup><sup> • </sup><sup>[8](https://www.ippp.dur.ac.uk/~krauss/Lectures/TheoreticalPhysics_2B/QM/Lecture15.pdf)</sup>.

The ladder must terminate at both ends, which forces \( \hat{J}^2|jm\rangle = j(j+1)\hbar^2 |jm\rangle \) with \( m \) running from \( -j \) to \( +j \) in integer steps, and \( j \) itself an integer or half-integer<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup><sup> • </sup><sup>[8](https://www.ippp.dur.ac.uk/~krauss/Lectures/TheoreticalPhysics_2B/QM/Lecture15.pdf)</sup>. The explicit matrix elements are

\[ \hat{J}_\pm |jm\rangle = \hbar \sqrt{(j \mp m)(j \pm m + 1)}\, |j, m \pm 1\rangle, \]

and in the standard basis \( \langle jm| \hat{J}^2 |jm'\rangle = j(j+1)\hbar^2 \delta_{mm'} \), \( \langle jm| \hat{J}_3 |jm'\rangle = m\hbar\, \delta_{mm'} \)<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>. Whether a given \( j \) occurs depends on the physical system: a spin-\( \frac{1}{2} \) particle has only \( j = \frac{1}{2} \), while a central-force problem admits only the integer orbital values<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>. Within one physical [Hilbert space](https://www.edgepedia.org/hilbert-space) the occurring \( j \) values are either all integer or all half-integer, never a mixture; combining states of the two kinds is excluded by a superselection rule<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>.

## Adding angular momenta

When a particle has both orbital and spin angular momentum, the total is \( \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} \), written more precisely as \( \hat{\mathbf{L}} \otimes \mathbb{I} + \mathbb{I} \otimes \hat{\mathbf{S}} \) because orbital and spin act on distinct Hilbert spaces<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup>. Since \( [\hat{L}_i, \hat{S}_j] = 0 \), the sum obeys the same algebra as its parts, so the whole eigenvalue structure carries over<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup>. The allowed total quantum numbers are bounded by the vector addition rule \( |l-s| \leq j \leq l+s \); for \( l = 1 \) and \( s = \frac{1}{2} \) the possibilities are \( j = \frac{1}{2} \) and \( j = \frac{3}{2} \)<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup>.

The coupled basis \( |j, m_j\rangle \) and the uncoupled product basis \( |l, m\rangle |s, m_s\rangle \) span the same space but are not the same states, because \( \hat{L}_z \) and \( \hat{S}_z \) do not commute with \( \hat{J}^2 \)<sup>[3](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)</sup>. The transformation coefficients between the two bases are the Clebsch-Gordan coefficients<sup>[12](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.07%3A_Adding_Angular_Momenta)</sup>. This bookkeeping has direct physical use: the spin-orbit interaction, a term proportional to \( \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \) in the hydrogen atom's Hamiltonian, is most naturally analyzed in the total-angular-momentum basis<sup>[12](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.07%3A_Adding_Angular_Momenta)</sup>.

## Representation theory

The commutation relations are the [Lie algebra](https://www.edgepedia.org/lie-algebra) su(2), and its irreducible matrix representations \( D_j \) are labeled by the spin \( j \), with basis \( \{|jm\rangle\} \) and dimension \( 2j+1 \)<sup>[7](https://www.physik.uni-leipzig.de/~rudolph/qm/qmgr.pdf)</sup>. In these representations \( \hat{J}_3 \) is diagonal with entries \( \hbar m \), while \( \hat{J}_1 \) and \( \hat{J}_2 \) connect adjacent \( m \) values with the square-root coefficients above<sup>[7](https://www.physik.uni-leipzig.de/~rudolph/qm/qmgr.pdf)</sup><sup> • </sup><sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>. Any operator that is a function of \( \hat{\mathbf{J}} \) alone does not mix subspaces of different \( j \), so its matrix in the standard basis decomposes into blocks of size \( 2j+1 \), one for each \( j \)<sup>[4](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)</sup>. This block structure is what makes the addition of angular momenta tractable: a composite system's representation space decomposes into a direct sum of irreducible spin-\( j \) subspaces, and the Clebsch-Gordan coefficients give the change of basis within each<sup>[12](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.07%3A_Adding_Angular_Momenta)</sup>.

## References

1. [Eyal Buks, Quantum Mechanics Lecture Notes, Ch. 6, Technion](https://buks.net.technion.ac.il/files/2024/10/qm_lecture_notes.pdf)
2. [Phys 506 Lecture 7: Rotations and Angular Momentum, Georgetown University](https://site.physics.georgetown.edu/~jkf/grad_quant2/html/lecture7.html)
3. [7.3: Total Angular Momentum, Advanced Quantum Mechanics (Kok), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_Mechanics_%28Kok%29/07%3A_Orbital_Angular_Momentum_and_Spin/7.03%3A_Total_Angular_Momentum)
4. [13. Representations of the Angular Momentum Operators and Rotations, Open Science Team Quantum Mechanics notes](https://openscienceteam.github.io/Quantum-Mechanics/repsamos/)
5. [The Feynman Lectures on Physics, Vol. III, Ch. 20: Operators](https://www.feynmanlectures.caltech.edu/III_20.html)
6. [Angular momentum and spin, Math foundations lecture notes](https://math.hodapp.com/lectures/physics/sicm-quantum-angular-momentum.html)
7. [Group Theoretical Aspects of Quantum Mechanics, Ch. 4, Universität Leipzig](https://www.physik.uni-leipzig.de/~rudolph/qm/qmgr.pdf)
8. [F. Krauss, Theoretical Physics II B, Lecture 15, IPPP Durham](https://www.ippp.dur.ac.uk/~krauss/Lectures/TheoreticalPhysics_2B/QM/Lecture15.pdf)
9. [4.1: Angular Momentum Operator Algebra, Quantum Mechanics (Fowler), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.01%3A_Angular_Momentum_Operator_Algebra)
10. [9.1: Spin Operators, Introductory Quantum Mechanics (Fitzpatrick), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/09%3A_Spin_Angular_Momentum/9.01%3A_Spin_Operators)
11. [4.2: Quantum Mechanics in 3D, Angular momentum, Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Introduction_to_Applied_Nuclear_Physics_%28Cappellaro%29/04%3A_Energy_Levels/4.02%3A_Quantum_Mechanics_in_3D_-_Angular_momentum)
12. [4.7: Adding Angular Momenta, Quantum Mechanics (Fowler), Physics LibreTexts](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Quantum_Mechanics_%28Fowler%29/04%3A_Angular_Momentum_Spin_and_the_Hydrogen_Atom/4.07%3A_Adding_Angular_Momenta)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Operators, observables, and angular momentum › Angular momentum operator algebra*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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