Short description: Quantum-mechanical operator associated with rotational symmetry

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In quantum mechanics, the angular momentum operator is the operator associated with rotational motion and rotational symmetry, and is the quantum analogue of angular momentum in classical mechanics. Classically, angular momentum is described by a vector $𝐋 = (L_{x}, L_{y}, L_{z})$ , whose components can all be specified simultaneously. In quantum mechanics, these components become operators ${\hat{L}}_{x}, {\hat{L}}_{y}, {\hat{L}}_{z}$ , representing measurements of angular momentum along each axis. However, unlike in classical physics, these operators do not commute, satisfying relations such as $[{\hat{L}}_{x}, {\hat{L}}_{y}] = i ℏ {\hat{L}}_{z}$ (and cyclic permutations), which implies that the components $L_{x}, L_{y}, L_{z}$ cannot be simultaneously known exactly. Instead, one can simultaneously determine the total angular momentum ${\hat{L}}^{2} = {\hat{L}}_{x}^{2} + {\hat{L}}_{y}^{2} + {\hat{L}}_{z}^{2}$ and a single component, usually ${\hat{L}}_{z}$ . Quantum states are therefore labeled by the quantum numbers $ℓ$ (total angular momentum) and $m$ (its projection along a chosen axis). Geometrically, this corresponds to a situation in which the magnitude of the angular momentum vector is well-defined while only one of its directional components is sharp, so the vector cannot be assigned a definite direction in three-dimensional space. Instead, it is often visualized as lying on a sphere of fixed radius (set by $L^{2}$ ) with uncertainty in its orientation, forming a “cone” of possible directions. As an observable, angular momentum is represented by operators whose eigenstates correspond to states with definite angular momentum, and whose eigenvalues give the possible results of measurement; it plays a central role in atomic physics, molecular physics, spectroscopy, and quantum theory more generally.

Angular momentum is one of the fundamental conserved quantities of motion, together with linear momentum and energy.^[1] In quantum mechanics, several related angular momentum operators appear:

the orbital angular momentum operator, usually denoted $𝐋$ ,
the spin angular momentum operator, usually denoted $𝐒$ ,
the total angular momentum operator, usually denoted $𝐉$ .

These are related by $𝐉 = 𝐋 + 𝐒 .$

Depending on context, the expression angular momentum operator may refer either to orbital angular momentum or to total angular momentum. For a closed system, the total angular momentum is conserved, in accordance with rotational symmetry and Noether's theorem.

Overview

In quantum mechanics, angular momentum appears in three closely related forms: orbital angular momentum, spin angular momentum, and total angular momentum.

Orbital angular momentum

The classical angular momentum of a particle is $𝐋 = 𝐫 \times 𝐩 .$ The same formal expression holds in quantum mechanics, except that $𝐫$ and $𝐩$ are now the position operator and momentum operator: $𝐋 = 𝐫 \times 𝐩 .$

Thus $𝐋$ is a vector operator, with components $𝐋 = (L_{x}, L_{y}, L_{z}) .$

For a single spinless, uncharged particle in the position representation, $𝐋 = - i ℏ (𝐫 \times \nabla),$ where $\nabla$ is the gradient operator.

Spin angular momentum

In addition to orbital angular momentum, quantum systems may possess an intrinsic form of angular momentum called spin, represented by $𝐒 = (S_{x}, S_{y}, S_{z}) .$

Spin has no exact classical counterpart. It is often illustrated as if a particle were spinning about an axis, but this picture is only heuristic; spin is an intrinsic quantum property.^[2] Elementary particles have fixed intrinsic spin: for example, electrons have spin $\frac{1}{2}$ , while photons have spin 1.

Total angular momentum

The total angular momentum operator combines orbital and spin contributions: $𝐉 = 𝐋 + 𝐒 .$

For a closed system, the total angular momentum is conserved. By contrast, $𝐋$ and $𝐒$ need not be conserved separately. For example, in spin–orbit interaction, angular momentum may be exchanged between orbital and spin parts while the total $𝐉$ remains constant.

Commutation relations

Commutation relations between components

The components of the orbital angular momentum operator satisfy the commutation relations^[3] $[L_{x}, L_{y}] = i ℏ L_{z}, [L_{y}, L_{z}] = i ℏ L_{x}, [L_{z}, L_{x}] = i ℏ L_{y},$ where the commutator is defined by $[X, Y] \equiv X Y - Y X .$

In index notation, $[L_{l}, L_{m}] = i ℏ \sum_{n = 1}^{3} ε_{l m n} L_{n},$ or, using Einstein summation convention, $[L_{l}, L_{m}] = i ℏ ε_{l m n} L_{n} .$

These relations can also be written compactly as^[4] $𝐋 \times 𝐋 = i ℏ 𝐋 .$

They follow from the canonical commutation relations $[x_{l}, p_{m}] = i ℏ δ_{l m} .$

The same algebra holds for spin and total angular momentum: $[S_{l}, S_{m}] = i ℏ \sum_{n = 1}^{3} ε_{l m n} S_{n}, [J_{l}, J_{m}] = i ℏ \sum_{n = 1}^{3} ε_{l m n} J_{n} .$

These commutation relations show that angular momentum operators generate the Lie algebra associated with three-dimensional rotations, usually written in physics as the algebra of SU(2) or SO(3).

Commutation relations involving the magnitude

The square of the orbital angular momentum operator is defined by $L^{2} \equiv L_{x}^{2} + L_{y}^{2} + L_{z}^{2} .$

This operator commutes with each component of $𝐋$ : $[L^{2}, L_{x}] = [L^{2}, L_{y}] = [L^{2}, L_{z}] = 0 .$

Thus one may simultaneously specify the values of $L^{2}$ and one chosen component, usually $L_{z}$ . The same property holds for spin and total angular momentum: $[S^{2}, S_{i}] = 0, [J^{2}, J_{i}] = 0 .$

Mathematically, $L^{2}$ is a Casimir invariant of the rotation algebra.

Uncertainty principle

Because different components of angular momentum do not commute, they cannot in general be measured simultaneously with arbitrary precision. For example, the Robertson–Schrödinger relation gives $σ_{L_{x}} σ_{L_{y}} \geq \frac{ℏ}{2} | ⟨ L_{z} ⟩ |,$ where $σ_{X}$ is the standard deviation of measurements of $X$ , and $⟨ X ⟩$ is the expectation value.

Thus two orthogonal components, such as $L_{x}$ and $L_{y}$ , are complementary observables. By contrast, $L^{2}$ and one component such as $L_{z}$ can be measured simultaneously.

Quantization

In quantum mechanics, angular momentum is quantized: only certain discrete measurement results are allowed. For orbital angular momentum,

Quantity	Allowed values	Notes
$L^{2}$	$ℏ^{2} ℓ (ℓ + 1)$ , where $ℓ = 0, 1, 2, \dots$	$ℓ$ is the orbital or azimuthal quantum number.
$L_{z}$	$ℏ m_{ℓ}$ , where $m_{ℓ} = - ℓ, - ℓ + 1, \dots, ℓ$	$m_{ℓ}$ is the magnetic quantum number.
$S^{2}$	$ℏ^{2} s (s + 1)$ , where $s = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots$	$s$ is the spin quantum number.
$S_{z}$	$ℏ m_{s}$ , where $m_{s} = - s, - s + 1, \dots, s$	$m_{s}$ is the spin projection quantum number.
$J^{2}$	$ℏ^{2} j (j + 1)$ , where $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots$	$j$ is the total angular momentum quantum number.
$J_{z}$	$ℏ m_{j}$ , where $m_{j} = - j, - j + 1, \dots, j$	$m_{j}$ is the total angular momentum projection quantum number.

For orbital angular momentum, both $ℓ$ and $m_{ℓ}$ are always integers. For spin and total angular momentum, half-integer values are also possible.

Derivation using ladder operators

A standard derivation of the allowed values uses the ladder operators $J_{+} \equiv J_{x} + i J_{y}, J_{-} \equiv J_{x} - i J_{y} .$

If $| ψ ⟩$ is a simultaneous eigenstate of $J^{2}$ and $J_{z}$ , then $J_{+} | ψ ⟩$ and $J_{-} | ψ ⟩$ are either zero or new simultaneous eigenstates with the same value of $J^{2}$ but with the $J_{z}$ eigenvalue shifted by $\pm ℏ$ . Repeated application of these operators leads to the quantization rules above.

Since $𝐋$ obeys the same algebra as $𝐉$ , the same ladder-operator argument applies to orbital angular momentum. In the case of $𝐋$ , single-valuedness of the wavefunction in the azimuthal angle $ϕ$ imposes the further restriction that $ℓ$ and $m_{ℓ}$ must be integers.

Visual interpretation

Although angular momentum in quantum mechanics is represented by operators rather than classical vectors, it is often illustrated heuristically by vectors of fixed length whose tip can lie only on a cone. For a state with given $ℓ$ and $m_{ℓ}$ , the magnitude is $| L | = \sqrt{L^{2}} = ℏ \sqrt{ℓ (ℓ + 1)},$ while the component $L_{z}$ has the definite value $L_{z} = ℏ m_{ℓ} .$

The uncertainty in the transverse components $L_{x}$ and $L_{y}$ is represented by the circular spread around the cone.

Quantization in macroscopic systems

The same quantum rules apply in principle to macroscopic bodies. In practice, however, the allowed steps in angular momentum are so small compared with the total angular momentum of ordinary objects that the spectrum appears continuous for all observable purposes.

Angular momentum as the generator of rotations

The most fundamental characterization of angular momentum is that it generates rotations.^[5] If $R (\hat{n}, ϕ)$ denotes the operator that rotates a system by an angle $ϕ$ about the axis $\hat{n}$ , then the angular momentum component along that axis is defined by $J_{\hat{n}} \equiv i ℏ \lim_{ϕ \to 0} \frac{R (\hat{n}, ϕ) - 1}{ϕ} = {i ℏ \frac{\partial R (\hat{n}, ϕ)}{\partial ϕ} |}_{ϕ = 0} .$

Equivalently, $R (\hat{n}, ϕ) = \exp (- \frac{i ϕ J_{\hat{n}}}{ℏ}) .$

Thus angular momentum governs how quantum states transform under rotations.

.

$R$ , associated with $𝐉$ , rotates the entire system.
$R_{spatial}$ , associated with $𝐋$ , rotates positions in space.
$R_{internal}$ , associated with $𝐒$ , rotates internal spin degrees of freedom.

]]

The orbital and spin operators similarly generate spatial and internal rotations: $R_{spatial} (\hat{n}, ϕ) = \exp (- \frac{i ϕ L_{\hat{n}}}{ℏ}),$ $R_{internal} (\hat{n}, ϕ) = \exp (- \frac{i ϕ S_{\hat{n}}}{ℏ}) .$

The relation $𝐉 = 𝐋 + 𝐒$ reflects the corresponding decomposition of a full rotation into spatial and internal parts.

SU(2), SO(3), and 360° rotations

In classical mechanics, a rotation by $36 0^{\circ}$ is identical to doing nothing. In quantum mechanics, however, a state with half-integer total angular momentum may change sign under a full $36 0^{\circ}$ rotation: $R (\hat{n}, 36 0^{\circ}) = - 1$ for half-integer $j$ , whereas $R (\hat{n}, 36 0^{\circ}) = + 1$ for integer $j$ .^[5]

This reflects the fact that quantum rotations are described by SU(2), which is the double cover of SO(3). Orbital angular momentum, by contrast, corresponds to ordinary spatial rotations and therefore only allows integer quantum numbers.

Connection to representation theory

When rotation operators act on quantum states, they define a representation of the rotation group. Correspondingly, angular momentum operators define a representation of the associated Lie algebra. The classification of possible angular momentum quantum numbers is therefore a representation-theoretic problem for SU(2) and SO(3).

Connection to commutation relations

Rotations about different axes do not commute. This noncommutativity is reflected at the operator level in the angular momentum commutation relations. Thus the algebra of angular momentum is a direct expression of the geometry of rotations in three-dimensional space.

Conservation of angular momentum

If the Hamiltonian $H$ is rotationally invariant, then angular momentum is conserved. Rotational invariance means $R H R^{- 1} = H,$ where $R$ is a rotation operator. This implies $[H, R] = 0,$ and therefore $[H, 𝐉] = 𝟎 .$

By the Ehrenfest theorem, the total angular momentum is then conserved. For a spinless particle in a central potential, this reduces to conservation of orbital angular momentum. When spin is present, spin–orbit coupling may transfer angular momentum between $𝐋$ and $𝐒$ , while the total $𝐉$ remains conserved.

Angular momentum coupling

When a system contains multiple sources of angular momentum, the individual contributions may combine to form a conserved total angular momentum. For example, for two angular momenta $𝐉_{1}$ and $𝐉_{2}$ , $𝐉 = 𝐉_{1} + 𝐉_{2} .$

The corresponding total quantum number satisfies $j \in {| j_{1} - j_{2} |, | j_{1} - j_{2} | + 1, \dots, j_{1} + j_{2}} .$

Transformations between uncoupled and coupled angular momentum bases are described by Clebsch–Gordan coefficients. In atomic and molecular physics, this structure underlies term symbols and the classification of energy levels.

Orbital angular momentum in spherical coordinates

Angular momentum operators naturally arise in problems with spherical symmetry. In the position representation and spherical coordinates, the orbital angular momentum operator is^[6] $\begin{aligned} 𝐋 & = i ℏ (\frac{\hat{θ}}{\sin (θ)} \frac{\partial}{\partial ϕ} - \hat{ϕ} \frac{\partial}{\partial θ}) \\ = i ℏ (\hat{𝐱} (\sin (ϕ) \frac{\partial}{\partial θ} + \cot (θ) \cos (ϕ) \frac{\partial}{\partial ϕ}) + \hat{𝐲} (- \cos (ϕ) \frac{\partial}{\partial θ} + \cot (θ) \sin (ϕ) \frac{\partial}{\partial ϕ}) - \hat{𝐳} \frac{\partial}{\partial ϕ}), \\ L_{+} & = ℏ e^{i ϕ} (\frac{\partial}{\partial θ} + i \cot (θ) \frac{\partial}{\partial ϕ}), \\ L_{-} & = ℏ e^{- i ϕ} (- \frac{\partial}{\partial θ} + i \cot (θ) \frac{\partial}{\partial ϕ}), \\ L^{2} & = - ℏ^{2} (\frac{1}{\sin (θ)} \frac{\partial}{\partial θ} (\sin (θ) \frac{\partial}{\partial θ}) + \frac{1}{\sin^{2} (θ)} \frac{\partial^{2}}{\partial ϕ^{2}}), \\ L_{z} & = - i ℏ \frac{\partial}{\partial ϕ} . \end{aligned}$

The angular part of the Laplace operator can be written in terms of $L^{2}$ : $Δ = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) - \frac{L^{2}}{ℏ^{2} r^{2}} .$

The simultaneous eigenstates of $L^{2}$ and $L_{z}$ satisfy $\begin{aligned} L^{2} | ℓ, m ⟩ & = ℏ^{2} ℓ (ℓ + 1) | ℓ, m ⟩, \\ L_{z} | ℓ, m ⟩ & = ℏ m | ℓ, m ⟩, \end{aligned}$ with wavefunctions $⟨ θ, ϕ | ℓ, m ⟩ = Y_{ℓ, m} (θ, ϕ),$ where $Y_{ℓ, m}$ are the spherical harmonics.^[7]