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Angular momentum operator

In quantum mechanics, the angular momentum operator is a fundamental vector operator \hat{\mathbf{L}} that quantifies the rotational dynamics of a quantum system, analogous to the classical angular momentum \mathbf{L} = \mathbf{r} \times \mathbf{p}. For orbital angular momentum, it is defined as \hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}, where \hat{\mathbf{r}} and \hat{\mathbf{p}} are the position and linear momentum operators, respectively, with components such as \hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y.^[1] The operator is Hermitian, ensuring real eigenvalues corresponding to measurable quantities, and its square \hat{L}^2 = \hat{\mathbf{L}} \cdot \hat{\mathbf{L}} represents the total magnitude.^[2] The components of the angular momentum operator satisfy the commutation relations [\hat{L}_x, \hat{L}_y] = i[\hbar](/page/H-bar) \hat{L}_z and cyclic permutations, which mirror the Lie algebra of the rotation group SO(3) and dictate the structure of angular momentum in quantum systems.^[1] Additionally, \hat{L}^2 commutes with each component, [\hat{L}^2, \hat{L}_i] = 0, allowing simultaneous eigenstates for \hat{L}^2 and one component (typically \hat{L}_z).^[2] The eigenvalues of \hat{L}^2 are [\hbar](/page/H-bar)^2 l(l+1) for integer quantum numbers l = 0, 1, 2, \dots, while those of \hat{L}_z are m[\hbar](/page/H-bar) with m = -l, -l+1, \dots, l.^[3] These relations arise from the operator definitions and the canonical commutation rules [\hat{x}_i, \hat{p}_j] = i[\hbar](/page/H-bar) \delta_{ij}.^[2] Beyond orbital angular momentum, the total angular momentum operator \hat{\mathbf{J}} = \hat{\mathbf{L}} + \hat{\mathbf{S}} incorporates intrinsic spin \hat{\mathbf{S}}, as in electrons or other particles with half-integer spin, and obeys identical commutation relations [\hat{J}_i, \hat{J}_j] = i[\hbar](/page/H-bar) \epsilon_{ijk} \hat{J}_k.^[3] Eigenvalues for \hat{J}^2 follow [\hbar](/page/H-bar)^2 j(j+1) where j can be integer or half-integer, enabling the addition of angular momenta via Clebsch-Gordan coefficients to couple multiple particles or subsystems.^[3] In systems with spherical symmetry, such as the hydrogen atom, the operators commute with the Hamiltonian, forming a complete set of commuting observables that simplify wavefunction separation into radial and angular parts.^[1] The angular momentum operators underpin key applications in quantum physics, including atomic spectra, molecular rotations, and particle physics, where they describe selection rules for transitions and symmetry under rotations.^[3] Ladder operators \hat{L}_\pm = \hat{L}_x \pm i \hat{L}_y facilitate raising and lowering m values, streamlining computations of matrix elements and spherical harmonics.^[2] Their abstract algebraic structure extends to other conserved quantities, like isospin in nuclear physics, highlighting their foundational role in understanding quantum symmetries.^[1]

Introduction

Definition and Classical Correspondence

In quantum mechanics, the angular momentum operator is defined as a vector operator \mathbf{L} = (L_x, L_y, L_z), whose components act on wave functions in the position representation.^[4] Specifically, the components are given by

L_i = -i\hbar (r_j \partial_k - r_k \partial_j),

where i, j, k are cyclic permutations of x, y, z, \hbar is the reduced Planck's constant, r_j and r_k are position coordinates, and \partial_j = \partial / \partial r_j are partial derivatives.^[4] This differential operator form arises naturally in the wave mechanics formulation, where it generates infinitesimal rotations in the Hilbert space of quantum states.^[4] Classically, angular momentum is the vector \mathbf{L} = \mathbf{r} \times \mathbf{p}, where \mathbf{r} is the position vector and \mathbf{p} is the linear momentum, representing the rotational analog of linear momentum in three-dimensional space.^[5] In the quantum mechanical promotion to operators, the position \mathbf{r} remains multiplication by the coordinate, while the momentum \mathbf{p} becomes -i\hbar \nabla, yielding \mathbf{L} = -i\hbar \mathbf{r} \times \nabla.^[5] This correspondence extends to the algebraic structure: classical Poisson brackets \{A, B\} between dynamical variables are replaced by commutators [A, B]/i\hbar in the quantum theory, ensuring that the Poisson bracket identity for angular momentum components translates to the corresponding quantum commutation relations.^[5] The angular momentum operator was introduced in the foundational development of matrix mechanics by Max Born, Werner Heisenberg, and Pascual Jordan in their 1926 paper, where it emerged as part of the systematic quantization of multi-degree-of-freedom systems, including derivations of conservation laws for angular momentum.^[6] This early formulation laid the groundwork for treating rotational dynamics quantum mechanically, bridging classical intuitions with the non-commutative operator algebra essential to the theory.^[6]

Role in Quantum Mechanics

The angular momentum operator plays a central role in quantum mechanics as the generator of infinitesimal rotations, embodying the principle that rotational symmetry leads to the conservation of angular momentum. According to the quantum analog of Noether's theorem, if the Hamiltonian of a system is invariant under rotations, the components of the angular momentum operator commute with the Hamiltonian, ensuring their expectation values remain constant over time.^[7] This conservation law underpins the analysis of isolated quantum systems, where rotational invariance—common in microscopic interactions—preserves total angular momentum, facilitating the classification of states and transitions. In atomic physics, the angular momentum operator is essential for interpreting phenomena like the Zeeman effect, in which atomic spectral lines split into multiple components under a weak magnetic field due to the torque exerted on the atom's total angular momentum by the field.^[8] Each energy level with total angular momentum quantum number J divides into $2J + 1 sublevels labeled by the magnetic quantum number M, with energy shifts proportional to M, enabling precise measurements of atomic structure and magnetic properties.^[8] In particle physics, the operator's intrinsic form, known as spin, connects directly to the spin-statistics theorem, which mandates that particles with half-integer spin values (fermions) follow antisymmetric wave functions and obey the Pauli exclusion principle, while those with integer spin (bosons) follow symmetric statistics.^[9] First rigorously proved by Pauli in 1940, this theorem explains the stability of matter by preventing fermions like electrons from occupying identical states, a cornerstone of quantum field theory and the standard model. Quantum information science exploits the angular momentum operator through spin-1/2 systems, where the two possible projections along an axis (\pm \hbar/2) encode the binary states of a qubit, enabling universal quantum computation via controlled manipulations of spin pairs.^[10] Unlike the linear momentum operator, which arises from translational symmetry and governs motion in uniform fields, the angular momentum operator addresses rotational dynamics, proving indispensable in central potentials like the Coulomb interaction in atoms, where it allows separation of radial and angular variables to yield exact solutions.^[11]

Components and Definitions

Orbital Angular Momentum

The orbital angular momentum operator \mathbf{L} for a single particle in quantum mechanics is defined analogously to its classical counterpart, as the cross product of the position operator \mathbf{r} and the linear momentum operator \mathbf{p}, yielding \mathbf{L} = \mathbf{r} \times \mathbf{p}.^[12] In the position representation, where \mathbf{p} = -i\hbar \nabla, this becomes the differential operator \mathbf{L} = -i\hbar \mathbf{r} \times \nabla.^[13] This form arises from the quantization of classical angular momentum, preserving the structure while incorporating the uncertainty principle through non-commuting operators.^[12] In spherical coordinates (r, \theta, \phi), the components of \mathbf{L} simplify significantly due to the rotational symmetry. The z-component, for instance, takes the form L_z = -i\hbar \frac{\partial}{\partial \phi}, which acts on a wavefunction \psi(r, \theta, \phi) by differentiating with respect to the azimuthal angle \phi.^[12] The full vector operator \mathbf{L} can be expressed using the gradient in spherical coordinates, \nabla = \hat{e}_r \frac{\partial}{\partial r} + \hat{e}_\theta \frac{1}{r} \frac{\partial}{\partial \theta} + \hat{e}_\phi \frac{1}{r \sin \theta} \frac{\partial}{\partial \phi}, leading to \mathbf{L} = -i\hbar \mathbf{r} \times \nabla.^[13] These expressions highlight that \mathbf{L} depends only on angular coordinates and derivatives, independent of the radial position r. The orbital angular momentum operator generates infinitesimal rotations in the particle's configuration space, effectively implementing rotations on scalar wavefunctions via the unitary transformation U(\hat{n}, \delta \theta) \psi(\mathbf{r}) = \psi(\mathbf{r} - \delta \theta \, \hat{n} \times \mathbf{r}) \approx [1 - i (\delta \theta / \hbar) \hat{n} \cdot \mathbf{L}] \psi(\mathbf{r}).^[12] This is particularly relevant for orbital motion in central force problems, such as the hydrogen atom or any spherically symmetric potential, where the Hamiltonian commutes with \mathbf{L}, conserving angular momentum.^[13] In these cases, the Schrödinger equation separates into radial and angular parts, with the angular portion resembling that of a rigid rotor constrained to a sphere.^[12] The eigenfunctions of the orbital angular momentum operators in the angular domain are the spherical harmonics Y_{lm}(\theta, \phi), which form a complete orthonormal basis for functions on the unit sphere and describe the quantized orbital states for such systems.^[13] These functions satisfy the operator equations without specifying the eigenvalue spectrum here, emphasizing their role in representing rotationally invariant solutions.^[12]

Spin Angular Momentum

The spin angular momentum operator \mathbf{S} represents an intrinsic property of elementary particles, distinct from orbital angular momentum, and is characterized by the quantum number s, which determines the possible eigenvalues of its magnitude squared operator \mathbf{S}^2 as s(s+1)\hbar^2, yielding a magnitude of \sqrt{s(s+1)}\hbar.^[14] For fundamental fermions like the electron, s = 1/2, while bosons such as photons have s = 1, and this intrinsic spin contributes to the particle's total angular momentum without relying on spatial coordinates or motion. The concept of electron spin was proposed in 1925 by George Uhlenbeck and Samuel Goudsmit to explain the fine structure splitting in atomic spectra, attributing it to an internal angular momentum of \hbar/2 for the electron, independent of its orbital motion.^[15] This proposal provided a theoretical framework that aligned with the anomalous Zeeman effect and other spectroscopic observations, resolving inconsistencies in earlier models.^[16] Experimentally, the idea gained support from the 1922 Stern-Gerlach experiment, which demonstrated the quantization of magnetic moments in silver atoms into two discrete states, later interpreted as evidence for spin-$1/2 projection eigenvalues \pm \hbar/2 along the measurement axis. For spin-$1/2 particles, the components of the spin operator are expressed as S_x = \frac{\hbar}{2} \sigma_x, S_y = \frac{\hbar}{2} \sigma_y, and S_z = \frac{\hbar}{2} \sigma_z, where the Pauli matrices \sigma_i (for i = x, y, z) are the 2×2 Hermitian, traceless matrices:

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.

^[14] These matrices satisfy \sigma_i^\dagger = \sigma_i, \operatorname{Tr}(\sigma_i) = 0, and \sigma_i^2 = I (the 2×2 identity), ensuring the spin operators are Hermitian and yield real eigenvalues, with the z-component eigenvalues \pm \hbar/2 corresponding to spin-up and spin-down states in the standard basis.^[17] This representation, introduced by Wolfgang Pauli in 1927, forms the basis for describing spin in non-relativistic quantum mechanics of magnetic electrons.^[14]

Total Angular Momentum

The total angular momentum operator \mathbf{J} in quantum mechanics is defined as the vector sum of the orbital angular momentum operator \mathbf{L} and the spin angular momentum operator \mathbf{S}, given by \mathbf{J} = \mathbf{L} + \mathbf{S}.^[3]^[18] This operator acts on the combined orbital-spin Hilbert space of a particle or system, where \mathbf{L} describes the rotational motion associated with the particle's position and momentum, while \mathbf{S} accounts for its intrinsic spin.^[19] The components of \mathbf{J} obey the same Lie algebra as those of \mathbf{L} or \mathbf{S} alone, satisfying the commutation relations [J_x, J_y] = i \hbar J_z and cyclic permutations, due to the fact that the orbital and spin operators commute, [\mathbf{L}, \mathbf{S}] = 0.^[20]^[3] This commutativity arises because \mathbf{L} acts solely on the spatial wave function and \mathbf{S} on the spinor part in the tensor product space. However, in the uncoupled basis of simultaneous eigenstates of \mathbf{L}^2, L_z, \mathbf{S}^2, and S_z, the components of \mathbf{J} mix the basis states, as J^2 does not commute with L_z or S_z individually, necessitating a transition to coupled representations for eigenstates of \mathbf{J}^2 and J_z.^[20]^[18] In atomic physics, the total angular momentum \mathbf{J} plays a central role in the fine structure of energy levels, where spin-orbit coupling splits degenerate states according to the value of j, the quantum number associated with \mathbf{J}, as seen in the hydrogen atom spectrum.^[19] Similarly, in nuclear physics, the total angular momentum quantum number J characterizes the spin and parity of nuclear states, determining the ordering and degeneracy of energy levels in nuclei through interactions that conserve total J.^[21] In multi-particle or atomic systems, coupled representations of \mathbf{J} are essential, often constructed via Clebsch-Gordan coefficients to handle the addition of multiple angular momenta.^[18]

Algebraic Structure

Operator Definitions in Cartesian Coordinates

In quantum mechanics, the angular momentum operators for orbital angular momentum are represented in the position basis using Cartesian coordinates as differential operators derived from the classical expression \mathbf{L} = \mathbf{r} \times \mathbf{p}, where \mathbf{p} = -i\hbar \nabla. The components take the explicit form

L_x = -i\hbar \left( y \frac{\partial}{\partial z} - z \frac{\partial}{\partial y} \right),

with cyclic permutations yielding

L_y = -i\hbar \left( z \frac{\partial}{\partial x} - x \frac{\partial}{\partial z} \right), \quad L_z = -i\hbar \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right).

These expressions act on wave functions \psi(\mathbf{r}) in three-dimensional space and correspond to the operator form L_i = \epsilon_{ijk} x_j p_k, where \epsilon_{ijk} is the Levi-Civita symbol and summation over repeated indices is implied.^[22]^[1] The operators L_x, L_y, and L_z are Hermitian, meaning \langle \psi | L_i | \phi \rangle = \langle L_i \psi | \phi \rangle for suitable wave functions \psi and \phi, ensuring real eigenvalues that correspond to measurable quantities. To verify this, consider the inner product \int \psi^* (L_x \phi) \, dV over all space. Substituting the differential form and applying integration by parts twice—first to transfer derivatives from \phi to \psi^*, then integrating by parts again—yields boundary terms that vanish for physically relevant wave functions (e.g., those decaying sufficiently fast at infinity). The result is \int (L_x \psi)^* \phi \, dV, confirming L_x^\dagger = L_x; the proof follows analogously for L_y and L_z.^[22] While the above definitions apply specifically to orbital angular momentum in position space, the angular momentum operators more generally—encompassing spin and total angular momentum—are defined abstractly as Hermitian operators satisfying the commutation relations [L_x, L_y] = i\hbar L_z (and cyclic permutations), without dependence on position or momentum coordinates. For spin, these operators act on an internal Hilbert space rather than configuration space, as exemplified by the Pauli matrices scaled by \hbar/2 for spin-1/2 particles.^[4]

Commutation Relations Among Components

The components of the angular momentum operator \mathbf{L} satisfy the canonical commutation relations [L_x, L_y] = i\hbar L_z, [L_y, L_z] = i\hbar L_x, and [L_z, L_x] = i\hbar L_y, which can be compactly written as [L_i, L_j] = i\hbar \epsilon_{ijk} L_k where \epsilon_{ijk} is the Levi-Civita symbol.^[2]^[1] These relations are derived from the definitions of the orbital angular momentum components in terms of position \mathbf{r} and momentum \mathbf{p} operators, L_x = y p_z - z p_y, L_y = z p_x - x p_z, and L_z = x p_y - y p_x, combined with the fundamental canonical commutation relations

[x_i, p_j] = i[\hbar](/page/H-bar) $$delta](/page/Delta)_{ij}

, [x_i, x_j] = [p_i, p_j] = 0.^[2]^[1] To compute, for example, [L_x, L_y], expand the commutator using the bilinearity properties: [L_x, L_y] = [y p_z, z p_x] - [y p_z, x p_z] - [z p_y, z p_x] + [z p_y, x p_z].^[2] Each term is then evaluated via the Leibniz (product) rule for commutators, [AB, C] = A[B, C] + [A, C]B and [A, BC] = B[A, C] + [A, B]C, applied to the monomial products in position and momentum operators, with repeated use of the canonical relations to simplify; the cross terms ultimately yield i[\hbar](/page/H-bar) (x p_y - y p_x) = i[\hbar](/page/H-bar) L_z.^[1] The cyclic permutations follow analogously. The same commutation algebra [J_i, J_j] = i\hbar \epsilon_{ijk} J_k holds universally for the spin angular momentum operators \mathbf{S}, which are postulated to satisfy these relations without a direct classical position-momentum origin, and for the total angular momentum \mathbf{J} = \mathbf{L} + \mathbf{S}, which inherits the algebra through the vector addition of operators obeying the same structure.^[23]^[1]

Magnitude and Its Commutators

The squared magnitude of the angular momentum operator is defined as [ \mathbf{L}^2 = L_x^2 + L_y^2 + L_z^2,

where $L_x$, $L_y$, and $L_z$ are the Cartesian components of the angular momentum operator.[](https://farside.ph.utexas.edu/teaching/qmech/Quantum/node71.html) This operator corresponds to the square of the total angular momentum vector in [quantum mechanics](/page/Quantum_mechanics).[](https://icourse.club/uploads/files/65fb4c16648d2b93f82fe3271c3381c211f2f532.pdf) A fundamental property of $\mathbf{L}^2$ is that it commutes with each component of the angular momentum operator:

[\mathbf{L}^2, L_i] = 0, \quad i = x, y, z.

This commutation relation holds due to the algebraic structure of the angular momentum operators.[](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/e765f050b2fd50a9d87d2ae801e9c52f_MIT8_05F13_Chap_09.pdf) To demonstrate it, consider the commutator $[\mathbf{L}^2, L_x]$. The term $[L_x^2, L_x]$ vanishes since any operator commutes with itself. For the remaining terms,

[L_y^2, L_x] = L_y [L_y, L_x] + [L_y, L_x] L_y = -i \hbar (L_y L_z + L_z L_y),

using the basic commutation relation $[L_x, L_y] = i \hbar L_z$ (and thus $[L_y, L_x] = -i \hbar L_z$). Similarly,

[L_z^2, L_x] = L_z [L_z, L_x] + [L_z, L_x] L_z = i \hbar (L_z L_y + L_y L_z),

with $[L_z, L_x] = i \hbar L_y$. Adding these contributions yields

[L_y^2 + L_z^2, L_x] = -i \hbar (L_y L_z + L_z L_y) + i \hbar (L_z L_y + L_y L_z) = 0,

since the expressions in parentheses are identical. The results for $[\mathbf{L}^2, L_y]$ and $[\mathbf{L}^2, L_z]$ follow by cyclic permutation.[](https://physicspages.com/pdf/Quantum%20mechanics/Angular%20momentum%20-%20commutators.pdf)[](https://icourse.club/uploads/files/65fb4c16648d2b93f82fe3271c3381c211f2f532.pdf) The commutation of $\mathbf{L}^2$ with all components implies that $\mathbf{L}^2$ and any single component, such as $L_z$, share a common eigenbasis, permitting the simultaneous measurement of the angular momentum magnitude squared and its projection along a chosen axis.[](https://physicspages.com/pdf/Quantum%20mechanics/Angular%20momentum%20-%20commutators.pdf) This property is central to the algebraic treatment of angular momentum in quantum systems with rotational symmetry.[](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/e765f050b2fd50a9d87d2ae801e9c52f_MIT8_05F13_Chap_09.pdf) The eigenvalues of $\mathbf{L}^2$ take the form $\hbar^2 l(l+1)$ for quantum number $l$, as explored in the quantization of angular momentum. ## Quantization and Eigenstates ### Eigenvalue Spectrum In quantum mechanics, the angular momentum operators $\mathbf{J}$ (encompassing both orbital $\mathbf{L}$ and spin $\mathbf{S}$ contributions) admit simultaneous eigenstates $|j, m\rangle$ for the squared magnitude $J^2$ and the $z$-component $J_z$, due to their commutation relation $[J^2, J_z] = 0$. These eigenstates satisfy the eigenvalue equations

J^2 |j, m\rangle = \hbar^2 j(j+1) |j, m\rangle, \quad J_z |j, m\rangle = \hbar m |j, m\rangle,

where the quantum number $j$ (often denoted $l$ for pure orbital angular momentum) takes non-negative values $j = 0, \frac{1}{2}, 1, \frac{3}{2}, \dots$, either integer for orbital cases or half-integer for spin contributions.[](https://people.chem.ucsb.edu/metiu/horia/OldFiles/QM2015/Ch13QM.pdf)[](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/e765f050b2fd50a9d87d2ae801e9c52f_MIT8_05F13_Chap_09.pdf) To derive these quantized eigenvalues, consider the operator identity $J^2 = J_z^2 + \frac{1}{2}(J_+ J_- + J_- J_+)$, where $J_\pm = J_x \pm i J_y$ are the raising and lowering combinations derived from the fundamental commutation relations $[J_x, J_y] = i \hbar J_z$ and cyclic permutations.[](https://people.chem.ucsb.edu/metiu/horia/OldFiles/QM2015/Ch13QM.pdf) Assuming a normalized state $|\psi\rangle$ that is an eigenvector of $J_z$ with eigenvalue $\hbar m$, the expectation value becomes $\langle J^2 \rangle = \hbar^2 m^2 + \frac{1}{2} \langle J_+ J_- + J_- J_+ \rangle$. The terms $\langle J_+ J_- \rangle = \|J_- |\psi\rangle\|^2 \geq 0$ and $\langle J_- J_+ \rangle = \|J_+ |\psi\rangle\|^2 \geq 0$ by the positivity of norms, yielding the inequality $\langle J^2 \rangle \geq \hbar^2 m^2$, or equivalently $j(j+1) \geq m^2$ upon identifying the eigenvalue of $J^2$ as $\hbar^2 j(j+1)$.[](https://people.chem.ucsb.edu/metiu/horia/OldFiles/QM2015/Ch13QM.pdf)[](http://physics.gmu.edu/~ellswort/p540/angmom.pdf) This bounds the possible $m$ values such that $|m| \leq j$, with $m$ differing by integers from $-j$ to $+j$. The specific form $j(j+1)$ (rather than $j^2$) arises from the algebraic closure of the representation under the action of $J_\pm$, ensuring a finite-dimensional spectrum.[](https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/e765f050b2fd50a9d87d2ae801e9c52f_MIT8_05F13_Chap_09.pdf) For a fixed $j$, the spectrum of $J_z$ exhibits $(2j + 1)$-fold degeneracy in the $m$ quantum number, as each $m$ corresponds to a distinct eigenvalue while sharing the same $J^2$ value; this degeneracy holds universally across systems, such as the [hydrogen atom](/page/Hydrogen_atom) where orbital angular momentum $l$ (integer $j$) labels subshells with $2l + 1$ states independent of the principal [quantum number](/page/Quantum_number) $n$.[](https://people.chem.ucsb.edu/metiu/horia/OldFiles/QM2015/Ch13QM.pdf)[](http://sites.science.oregonstate.edu/~tatej/COURSES/ph426/qmch8.pdf) ### Ladder Operator Method The ladder operator method provides an algebraic framework for determining the eigenvalues and eigenstates of the [angular momentum](/page/Angular_momentum) operators by exploiting the commutation relations to construct [raising](/page/Raising) and lowering operators. These operators systematically connect states within the same multiplet, revealing the [discrete](/page/Discrete) spectrum without solving differential equations.[](https://quantummechanics.ucsd.edu/ph130a/130_notes/node209.html) Define the [raising](/page/Raising) and lowering operators as $ L_+ = L_x + i L_y $ and $ L_- = L_x - i L_y $, respectively. These satisfy the commutation relations $[L_z, L_\pm] = \pm \hbar L_\pm$ and $[L^2, L_\pm] = 0$, ensuring that $L_\pm$ map eigenstates of $L^2$ and $L_z$ to other eigenstates within the same $L^2$ eigenspace but shifted in the $L_z$ eigenvalue. Additionally, the identity $L^2 - L_z^2 = \frac{1}{2} (L_+ L_- + L_- L_+)$ follows from expressing the Cartesian components in terms of $L_\pm$.[](https://www.andrew.cmu.edu/user/suter/AngMomentum.pdf)[](https://quantummechanics.ucsd.edu/ph130a/130_notes/node209.html) Consider a simultaneous eigenstate $|l, m\rangle$ of $L^2$ and $L_z$, satisfying $L^2 |l, m\rangle = [\hbar](/page/H-bar)^2 l(l+1) |l, m\rangle$ and $L_z |l, m\rangle = [\hbar](/page/H-bar) m |l, m\rangle$, where $l$ and $m$ are to be determined. Applying $L_+$ yields $L_z (L_+ |l, m\rangle) = L_+ (L_z + [\hbar](/page/H-bar)) |l, m\rangle = [\hbar](/page/H-bar) (m + 1) (L_+ |l, m\rangle)$, so $L_+ |l, m\rangle$ is an eigenstate of $L_z$ with eigenvalue $[\hbar](/page/H-bar) (m + 1)$. Since $[L^2, L_+] = 0$, it shares the same $L^2$ eigenvalue $[\hbar](/page/H-bar)^2 l(l+1)$. Similarly, $L_- |l, m\rangle$ connects to the state with $m - 1$. The states form a ladder, with $L_+$ raising $m$ by 1 and $L_-$ lowering it by 1.[](https://www.andrew.cmu.edu/user/suter/AngMomentum.pdf) To find the action explicitly, compute the norm $\|L_\pm |l, m\rangle\|^2 = \langle l, m | L_\mp L_\pm | l, m \rangle$. Using the identity for $L^2 - L_z^2$ and the eigenvalues, this yields $\|L_+ |l, m\rangle\|^2 = \hbar^2 [l(l+1) - m(m+1)]$. Thus, the normalized action is $L_+ |l, m\rangle = \hbar \sqrt{l(l+1) - m(m+1)} \, |l, m+1\rangle$, and analogously $L_- |l, m\rangle = \hbar \sqrt{l(l+1) - m(m-1)} \, |l, m-1\rangle$. The matrix elements are $\langle l, m' | L_\pm | l, m \rangle = \hbar \sqrt{l(l+1) - m m'} \, \delta_{m', m \pm 1}$.[](https://quantummechanics.ucsd.edu/ph130a/130_notes/node209.html) The ladder must be finite because the norms must be non-negative. Applying $L_+$ repeatedly increases $m$ until $l(l+1) - m(m+1) = 0$ for some maximum $m_{\max} = l$, where $L_+ |l, l\rangle = 0$. Similarly, lowering reaches $m_{\min} = -l$, with $L_- |l, -l\rangle = 0$. For the ladder to close consistently, the number of states is $2l + 1$, and applying the norm condition at $m = l$ gives $l(l+1) - l(l+1) = 0$, confirming the eigenvalue of $L^2$. Both $l$ and $m$ take values such that $m$ ranges from $-l$ to $l$ in integer steps, with $l = 0, 1/2, 1, 3/2, \dots$. This algebraic procedure establishes the quantized spectrum and the structure of the eigenstates.[](https://www.andrew.cmu.edu/user/suter/AngMomentum.pdf)[](https://quantummechanics.ucsd.edu/ph130a/130_notes/node209.html) ### Spherical Harmonics for Orbital Case In the orbital angular momentum case, the simultaneous eigenfunctions of the operators $ \hat{L}^2 $ and $ \hat{L}_z $ in position space are the [spherical harmonics](/page/Spherical_harmonics) $ Y_l^m(\theta, \phi) $, where $ l $ is a non-negative integer and $ m = -l, -l+1, \dots, l $.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/orbamsph.pdf) These functions satisfy the eigenvalue equations

\hat{L}^2 Y_l^m(\theta, \phi) = \hbar^2 l(l+1) Y_l^m(\theta, \phi), \quad \hat{L}_z Y_l^m(\theta, \phi) = \hbar m Y_l^m(\theta, \phi),

providing the basis for expanding angular-dependent wave functions in [quantum mechanics](/page/Quantum_mechanics).[31] The spherical harmonics take the explicit form

Y_l^m(\theta, \phi) = (-1)^m \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}} P_l^m(\cos\theta) , e^{i m \phi}

for $ m \geq 0 $, with the associated Legendre functions $ P_l^m(\xi) $ defined as

P_l^m(\xi) = (1 - \xi^2)^{m/2} \frac{d^m}{d\xi^m} P_l(\xi),

where $ P_l(\xi) $ are the [Legendre polynomials](/page/Legendre_polynomials) and $ \xi = \cos\theta $.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/orbamsph.pdf) For $ m < 0 $, $ Y_l^m(\theta, \phi) = (-1)^m Y_l^{-m *}(\theta, \phi) $.[](https://farside.ph.utexas.edu/teaching/qm/lectures/node40.html) The associated Legendre functions arise from solving the angular part of the Laplace equation in spherical coordinates, ensuring the separation of variables for the eigenproblem.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/orbamsph.pdf) The [spherical harmonics](/page/Spherical_harmonics) form a complete [orthonormal basis](/page/Orthonormal_basis) on the unit sphere, satisfying the orthogonality relation

\int Y_l^{m *}(\theta, \phi) Y_{l'}^{m'}(\theta, \phi) , d\Omega = \delta_{l l'} \delta_{m m'},

where the integral is over the [solid angle](/page/Solid_angle) $ d\Omega = \sin\theta \, d\theta \, d\phi $ with limits $ \theta $ from 0 to $ \pi $ and $ \phi $ from 0 to $ 2\pi $.[](https://farside.ph.utexas.edu/teaching/qm/lectures/node40.html) This normalization, $ \int |Y_l^m|^2 d\Omega = 1 $, facilitates the expansion of any [square-integrable function](/page/Square-integrable_function) on the sphere.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/orbamsph.pdf) Under parity transformation (spatial inversion), the spherical harmonics transform as $ Y_l^m(\pi - \theta, \phi + \pi) = (-1)^l Y_l^m(\theta, \phi) $, a property that determines the parity of orbital states and influences selection rules for electric dipole transitions in atomic physics.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/orbamsph.pdf) ## Symmetry and Rotations ### Generator of Rotations In quantum mechanics, the angular momentum operator $\mathbf{J}$ serves as the generator of rotations, meaning that unitary transformations corresponding to spatial rotations of the system are expressed in terms of exponentials involving $\mathbf{J}$. Specifically, the rotation operator for a rotation by an angle $\theta$ about a unit vector $\hat{\mathbf{n}}$ is given by

U(\hat{\mathbf{n}}, \theta) = e^{-i \theta \mathbf{J} \cdot \hat{\mathbf{n}} / \hbar},

where $\hbar$ is the reduced Planck's constant. This operator acts on the Hilbert space of quantum states, implementing the rotation in a manner consistent with the principles of quantum symmetry transformations.[](https://bohr.physics.berkeley.edu/classes/221/1112/notes/spinrot.pdf)[](https://farside.ph.utexas.edu/teaching/qm/lectures/node39.html) For infinitesimal rotations, where $\theta$ is small, the rotation operator expands via the [Taylor series](/page/Taylor_series) to

\delta U \approx 1 - i (\theta / \hbar) \mathbf{J} \cdot \hat{\mathbf{n}},

\frac{d}{dt} \langle \mathbf{L} \rangle = \frac{i}{\hbar} \langle [H, \mathbf{L}] \rangle,

where $H$ is the Hamiltonian and $[ \cdot, \cdot ]$ denotes the commutator. For a rotationally invariant Hamiltonian, which depends only on scalar combinations of coordinates and momenta (such as $r = |\mathbf{r}|$ in central potentials), the components of $\mathbf{L}$ commute with $H$, yielding $[H, \mathbf{L}] = 0$. Consequently, $\frac{d}{dt} \langle \mathbf{L} \rangle = 0$, ensuring that the expectation value of angular momentum remains constant over time.[](https://www.physics.rutgers.edu/~steves/Mark_and_Janos/Lectures/Lecture_9_Ehrenfest_Theorem.pdf)[](https://icourse.club/uploads/files/65fb4c16648d2b93f82fe3271c3381c211f2f532.pdf) This conservation extends to the total [angular momentum](/page/Angular_momentum) $\mathbf{J} = \mathbf{L} + \mathbf{S}$ in isolated [quantum systems](/page/Quantum-Systems) lacking external torques, such as those without [magnetic fields](/page/The_Magnetic_Fields) or asymmetric potentials. In such cases, the [Hamiltonian](/page/Hamiltonian) commutes with $\mathbf{J}$, $[H, \mathbf{J}] = 0$, preserving both the [magnitude](/page/Magnitude) $J^2$ and a chosen component (e.g., $J_z$) of the total angular momentum. A representative example occurs in atomic collisions, where two neutral atoms interact via short-range forces in an otherwise [empty space](/page/The_Empty_Space); the total angular momentum of the colliding pair, including initial orbital contributions from their relative motion, remains conserved throughout the [scattering](/page/Scattering) process, dictating selection rules for final states and angular distributions.[](https://icourse.club/uploads/files/65fb4c16648d2b93f82fe3271c3381c211f2f532.pdf) The underlying principle is the quantum analog of [Noether's theorem](/page/Noether's_theorem), which associates continuous symmetries of the [Hamiltonian](/page/Hamiltonian) with conserved quantities through operator commutators. Rotational invariance implies that the generators of rotations—the angular momentum components—satisfy $[H, J_i] = 0$ for $i = x, y, z$, establishing them as conserved charges without explicit time dependence in the [equations of motion](/page/Equations_of_motion). This framework highlights how spatial isotropy enforces [angular momentum](/page/Angular_momentum) conservation, mirroring classical results but realized via unitary representations of the rotation group.[](https://icourse.club/uploads/files/65fb4c16648d2b93f82fe3271c3381c211f2f532.pdf)[](https://www.feynmanlectures.caltech.edu/III_17.html) ### Coupling of Multiple Angular Momenta In [quantum mechanics](/page/Quantum_mechanics), the coupling of multiple angular momenta arises when combining the angular momentum operators of two or more subsystems, such as electrons or particles, to form a total angular momentum operator $\mathbf{J} = \sum_i \mathbf{J}_i$. For two angular momenta $\mathbf{J}_1$ and $\mathbf{J}_2$ with quantum numbers $j_1$ and $j_2$, the possible values of the total [angular momentum](/page/Angular_momentum) quantum number $j$ range from $|j_1 - j_2|$ to $j_1 + j_2$ in steps of 1, reflecting the decomposition of the [tensor product](/page/Tensor_product) of irreducible representations of the rotation group. This addition rule ensures that the total $m = m_1 + m_2$ is conserved, with each $j$ subspace being degenerate in $m$ from $-j$ to $j$. The coupled basis states, denoted $|j, m; j_1 j_2\rangle$, diagonalize $\mathbf{J}^2$ and $J_z$, and are expressed as linear combinations of the [uncoupled](/page/Uncoupled) product states $|j_1 m_1\rangle |j_2 m_2\rangle$:

|j, m; j_1 j_2\rangle = \sum_{m_1, m_2} \langle j_1 m_1 j_2 m_2 | j m \rangle , |j_1 m_1\rangle |j_2 m_2\rangle,

where the expansion coefficients $\langle j_1 m_1 j_2 m_2 | j m \rangle$ are the Clebsch-Gordan coefficients, real numbers that vanish unless $m = m_1 + m_2$. These coefficients satisfy [orthogonality](/page/Orthogonality) relations, such as $\sum_{m} \langle j_1 m_1 j_2 m_2 | j m \rangle \langle j' m | j_1 m_1 j_2 m_2 \rangle = \delta_{j j'}$, ensuring the basis transformation is unitary. They can be computed analytically for general $j_1, j_2$ using recursive formulas or explicit expressions involving factorials and binomial coefficients, though tables are commonly used for practical calculations. For low values of $j_1$ and $j_2$, the Clebsch-Gordan coefficients take simple forms that facilitate applications like spin-orbit coupling in atoms, where $\mathbf{J} = \mathbf{L} + \mathbf{S}$ with $j_2 = 1/2$. For instance, when $j_1 = l$ (orbital angular momentum) and $j_2 = 1/2$, the states are:

\left| j = l + \frac{1}{2}, m \right\rangle = \sqrt{\frac{l + m + 1/2}{2l + 1}} , |l, m - 1/2\rangle , \left|\frac{1}{2}, \frac{1}{2}\right\rangle + \sqrt{\frac{l - m + 1/2}{2l + 1}} , |l, m + 1/2\rangle , \left|\frac{1}{2}, -\frac{1}{2}\right\rangle,

\left| j = l - \frac{1}{2}, m \right\rangle = -\sqrt{\frac{l - m + 1/2}{2l + 1}} , |l, m - 1/2\rangle , \left|\frac{1}{2}, \frac{1}{2}\right\rangle + \sqrt{\frac{l + m + 1/2}{2l + 1}} , |l, m + 1/2\rangle , \left|\frac{1}{2}, -\frac{1}{2}\right\rangle,

with coefficients derived from the general [orthogonality](/page/Orthogonality) and [normalization](/page/Normalization) conditions. These explicit values are tabulated for small $j$ up to 5/2 in standard references, enabling straightforward computation of [matrix](/page/Matrix) elements in [perturbation theory](/page/Perturbation_theory) or fine-structure calculations. For systems involving three or more angular momenta, such as multi-electron atoms, [direct coupling](/page/Direct_coupling) becomes cumbersome, necessitating recoupling schemes to transform between different pairing conventions, like $((j_1 j_2) j_{12} j_3) j$ and $(j_1 (j_2 j_3) j_{23}) j$. This recoupling is quantified by the Wigner 6j symbols $\begin{Bmatrix} j_1 & j_2 & j_{12} \\ j_3 & j & j_{23} \end{Bmatrix}$, which appear in the overlap $\langle (j_1 j_2)_{j_{12}} j_3 ; j m | j_1 (j_2 j_3)_{j_{23}} ; j m \rangle = (-1)^{\phi} \sqrt{(2j_{12}+1)(2j_{23}+1)} \begin{Bmatrix} j_1 & j_2 & j_{12} \\ j_3 & j & j_{23} \end{Bmatrix}$, where $\phi$ is a [phase factor](/page/Phase_factor). The 6j symbols, introduced by Wigner and systematized by Racah, satisfy symmetry properties and sum rules that simplify multi-angular momentum calculations. For four or more momenta, 9j symbols extend this framework, relating three possible coupling paths. In [atomic physics](/page/Atomic_physics), recoupling via 6j and 9j symbols is essential for comparing [coupling](/page/Coupling) schemes in multi-electron systems. In LS (Russell-Saunders) [coupling](/page/Coupling), individual orbital angular momenta $\mathbf{l}_i$ couple to total $\mathbf{L} = \sum \mathbf{l}_i$ and spins to $\mathbf{S} = \sum \mathbf{s}_i$, then $\mathbf{J} = \mathbf{L} + \mathbf{S}$, suitable for light atoms where electrostatic interactions [dominate](/page/Dominate) spin-orbit effects. Conversely, jj [coupling](/page/Coupling) pairs each electron's $\mathbf{j}_i = \mathbf{l}_i + \mathbf{s}_i$ before summing to total $\mathbf{J}$, prevailing in heavy atoms with strong spin-orbit [coupling](/page/Coupling); intermediate cases use recoupling coefficients to interpolate between schemes. These transformations, computed using 6j symbols, allow consistent [term symbol](/page/Term_symbol) assignments and energy level predictions across the periodic table. ## Coordinate Representations ### Orbital Angular Momentum in Spherical Coordinates In spherical coordinates $(r, \theta, \phi)$, the orbital angular momentum operator $\mathbf{L} = \mathbf{r} \times \mathbf{p}$ takes the form of differential operators acting on the angular variables $\theta$ and $\phi$, while being independent of the radial coordinate $r$. The components are expressed as follows:

L_x = -i\hbar \left( -\sin\phi \frac{\partial}{\partial \theta} - \cot\theta \cos\phi \frac{\partial}{\partial \phi} \right),

L_y = -i\hbar \left( \cos\phi \frac{\partial}{\partial \theta} - \cot\theta \sin\phi \frac{\partial}{\partial \phi} \right),

L_z = -i\hbar \frac{\partial}{\partial \phi},

and the magnitude squared is

L^2 = -\hbar^2 \left[ \frac{1}{\sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial \phi^2} \right].

These operators satisfy the commutation relations $[L_x, L_y] = i\hbar L_z$ and cyclic permutations, generating infinitesimal rotations in the angular domain.[](https://www.asc.ohio-state.edu/jayaprakash.1/828/828ho2.pdf) For a particle in a central potential $V(r)$, the time-independent Schrödinger equation $\left[ -\frac{\hbar^2}{2\mu} \nabla^2 + V(r) \right] \psi = E \psi$ separates in spherical coordinates due to the form of the Laplacian $\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right) - \frac{L^2}{r^2 \hbar^2}$. Assuming a separable wave function $\psi(r, \theta, \phi) = R(r) Y(\theta, \phi)$, the angular part yields the eigenvalue equation $L^2 Y = \hbar^2 l(l+1) Y$ with $l = 0, 1, 2, \dots$, while $L_z Y = \hbar m Y$ with $m = -l, \dots, l$. The solutions $Y_{l}^m(\theta, \phi)$ are the spherical harmonics. This separation reduces the problem to a radial equation for $R(r)$ and an angular one solved by the angular momentum operators.[](https://bohr.physics.berkeley.edu/classes/221/notes/orbamsph.pdf) In the semiclassical limit of large quantum numbers, the quantization of orbital [angular momentum](/page/Angular_momentum) finds analogy in rotating macroscopic systems, such as diatomic molecules modeled as [rigid rotors](/page/Rigid_rotor). Here, the Bohr-Sommerfeld quantization condition $\oint p_\phi d\phi = n h$ (with integer $n$) leads to discrete rotational energy levels $E \approx \frac{n^2 \hbar^2}{2I}$ (where $I$ is the [moment of inertia](/page/Moment_of_inertia)), approximating the quantum [rigid rotor](/page/Rigid_rotor) spectrum $E_J = \frac{J(J+1) \hbar^2}{2I}$ for large $J$, as originally applied by Ehrenfest in [1913](/page/1913) using adiabatic invariance.[](https://philsci-archive.pitt.edu/16989/1/Quantum-handbook.pdf) ### Matrix Representations for Spin The matrix representations of the spin angular momentum operators $\mathbf{S}$ are constructed in the Hilbert space of dimension $2s + 1$, where $s$ is the spin quantum number, using the standard basis states $|s, m_s\rangle$ with $m_s = -s, -s+1, \dots, s$. These representations satisfy the commutation relations $[S_x, S_y] = i \hbar S_z$ and cyclic permutations, as derived from the Lie algebra of SU(2).[](https://mpl.mpg.de/fileadmin/user_upload/Marquardt_Division/Teaching/Angular_Momentum.pdf) For the simplest case of spin-$s = 1/2$ (e.g., electrons or other spin-1/2 particles), the operators are proportional to the Pauli matrices $\boldsymbol{\sigma}$, given explicitly as:

S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \ 1 & 0 \end{pmatrix}, \quad S_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \ i & 0 \end{pmatrix}, \quad S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix},

in the basis where $|1/2, 1/2\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $|1/2, -1/2\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. These matrices were introduced by Pauli to describe the two-valuedness of electron spin and form the fundamental representation of SU(2).[](https://arxiv.org/pdf/2304.03139.pdf) In general, for arbitrary spin $s$, the operators $S_z$ is diagonal in the $|s, m_s\rangle$ basis:

\langle s, m_s' | S_z | s, m_s \rangle = m_s \hbar , \delta_{m_s', m_s},

while the raising and lowering operators are $S_+ = S_x + i S_y$ and $S_- = S_x - i S_y$, with matrix elements

\langle s, m_s' | S_+ | s, m_s \rangle = \hbar \sqrt{(s - m_s)(s + m_s + 1)} , \delta_{m_s', m_s + 1},

\langle s, m_s' | S_- | s, m_s \rangle = \hbar \sqrt{(s + m_s)(s - m_s + 1)} , \delta_{m_s', m_s - 1}.

The transverse components follow as $S_x = (S_+ + S_-)/2$ and $S_y = (S_+ - S_-)/(2i)$, yielding tridiagonal matrices for $S_x$ and $S_y$. These elements ensure the total spin operator satisfies $S^2 |s, m_s\rangle = \hbar^2 s(s+1) |s, m_s\rangle$.[](https://mpl.mpg.de/fileadmin/user_upload/Marquardt_Division/Teaching/Angular_Momentum.pdf) For spin-$s=1$ (e.g., vector bosons like photons in certain contexts), the matrices in the basis $|1,1\rangle, |1,0\rangle, |1,-1\rangle$ are:

S_x = \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 1 \ 0 & 1 & 0 \end{pmatrix}, \quad S_y = \frac{\hbar}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \ i & 0 & -i \ 0 & i & 0 \end{pmatrix}, \quad S_z = \hbar \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & -1 \end{pmatrix}.

Higher-spin representations follow the same ladder operator construction, with increasing matrix size and sparsity, and are irreducible under SU(2) transformations. These forms are essential for computing [expectation](/page/Expectation) values, [time evolution](/page/Time_evolution), and entanglement in multi-spin systems.[50]

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