Angular momentum operator

The vector operator whose components generate rotations and represent angular momentum in a quantum system. Its components obey angular-momentum commutation relations and have quantized spectra.

Angular momentum commutation relations: The commutators among angular momentum components, including [Jx, Jy] = iℏJz and cyclic permutations. They encode the operator’s rotational structure and prevent all three components from having definite values together.
Observable (quantum mechanics): A measurable physical quantity represented in quantum theory by a Hermitian operator. Angular momentum is represented by operators whose eigenvalues are possible measurement outcomes.
Addition of angular momentum: The quantum-mechanical procedure for combining angular momenta into total-angular-momentum states. Tensor products of angular-momentum operators describe composite systems and coupled particles.
Classical angular momentum: The vector r × p describing rotational motion in classical mechanics. The quantum operator preserves the classical quantity’s role while adding noncommuting components and discrete spectra.
Angular momentum ladder operators: The operators J+ and J− that raise or lower an angular-momentum state’s magnetic quantum number. They generate the allowed states within a fixed total-angular-momentum multiplet.
Hilbert space: A complete inner-product space used to represent quantum states and linear operators. Angular momentum operators act on state vectors in a quantum system’s Hilbert space.
Clebsch–Gordan coefficients: Coefficients that transform between uncoupled and coupled bases for two angular momenta. They express total-angular-momentum eigenstates using individual subsystem states.
Orbital angular momentum: The angular momentum associated with a particle’s spatial motion, represented by r × p in quantum mechanics. It contrasts with intrinsic spin, which is not generated by position and momentum.
Angular momentum eigenstates: Quantum states with definite eigenvalues of total angular momentum squared and one chosen component. They are the states that diagonalize the operator’s compatible observables J² and Jz.
Hermitian operator: A linear operator equal to its adjoint, with real eigenvalues and orthogonal eigenvectors under suitable conditions. Each measurable angular-momentum component is represented by a Hermitian operator.

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