Quantum Angular Momentum
Angular momentum in quantum mechanics is not merely the quantization of $\mathbf{L}=\mathbf{r}\times\mathbf{p}$. That formula describes one specific realization (orbital angular momentum). The deeper unifying idea is that angular momentum is the generator of rotations acting on quantum states in Hilbert space, and orbital angular momentum, spin, and total angular momentum are different representations of the same rotational structure.
Angular momentum is the generator of rotations in Hilbert space.
Orbital angular momentum, spin, and total angular momentum are different representations of this same rotational structure.
Generators and the Rotation Algebra
In classical mechanics, an infinitesimal transformation of an observable $f$ generated by $G$ is written using the Poisson bracket:
\[\delta f=\varepsilon\{f,G\}.\]For spatial rotations, the classical generator is angular momentum; for orbital motion,
\[\mathbf{L}=\mathbf{r}\times\mathbf{p}.\]In quantum mechanics, the classical bracket-based notion of generation is replaced by commutators:
\[\{A,B\}\longrightarrow \frac{1}{i\hbar}[A,B].\]A small rotation by angle $d\phi$ about the axis $\hat n$ is represented by a unitary operator
\[U(\hat n,d\phi)=1-\frac{i}{\hbar}(\mathbf{J}\cdot\hat n)\,d\phi,\]so $\mathbf{J}$ is defined operationally as the generator of rotations on states.
Classically, angular momentum is often visualized as rotating matter.
Quantum mechanically, angular momentum is first defined as the operator that generates rotation of the state.
Rotations about different axes do not commute. The group structure of 3D rotations forces the generators to satisfy
\[[J_x,J_y]=i\hbar J_z,\] \[[J_y,J_z]=i\hbar J_x,\] \[[J_z,J_x]=i\hbar J_y.\]Equivalently, in compact tensor form,
\[[J_i,J_j]=i\hbar\,\varepsilon_{ijk}J_k.\]The non-commutativity of $J_x$, $J_y$, and $J_z$ is the algebraic shadow of the non-commutativity of rotations in three-dimensional space.
1. rotate slightly about $x$,
2. rotate slightly about $y$,
3. undo the $x$-rotation,
4. undo the $y$-rotation.
If rotations commuted, this closed operation would give the identity: $$ K=1. $$ But 3D rotations do not commute, so a small residual rotation remains. Since the residual effect is produced by mixing an $x$-rotation and a $y$-rotation, the right-hand rule says the leftover rotation is about the $z$-axis. Therefore the closed operation has the form $$ K=1-\frac{i}{\hbar}J_z\,d\gamma. $$ The angle of this residual rotation is second order, because it appears only from the combined effect of the two small rotations: $$ d\gamma \sim d\alpha d\beta. $$ With the chosen convention, $$ d\gamma=d\alpha d\beta. $$ Hence $$ K=U_z(d\alpha d\beta)=1-\frac{i}{\hbar}J_z\,d\alpha d\beta. $$ So the intuitive meaning is: A tiny $x$-rotation followed around a tiny $y$-rotation loop leaves behind a tiny $z$-rotation. That is why the algebra becomes $$ [J_x,J_y]\propto J_z. $$ The commutator of the generators remembers the small leftover rotation produced by the non-commutativity of the rotation group.
Eigenvalues, $J^2$ and $J_z$, and Ladder Structure
Because $J_x$, $J_y$, and $J_z$ do not commute, they cannot be simultaneously sharp. The operator
\[J^2=J_x^2+J_y^2+J_z^2\]commutes with each component; in particular,
\[[J^2,J_z]=0.\]Hence one classifies states by simultaneous eigenvalues of $J^2$ and $J_z$:
\[J^2|j,m\rangle=j(j+1)\hbar^2|j,m\rangle,\] \[J_z|j,m\rangle=m\hbar|j,m\rangle.\]Here $j$ labels the multiplet (total angular momentum) and $m$ labels the projection along a chosen axis.
A classical angular momentum vector can have all three components specified.
A quantum angular momentum state is classified by $j$ and a single chosen projection $m$.
Define ladder operators
\[J_+=J_x+iJ_y,\] \[J_-=J_x-iJ_y.\]They satisfy
\[[J_z,J_+]=\hbar J_+,\] \[[J_z,J_-]=-\hbar J_-,\]and act as
\[J_+|j,m\rangle=\hbar\sqrt{(j-m)(j+m+1)}|j,m+1\rangle,\] \[J_-|j,m\rangle=\hbar\sqrt{(j+m)(j-m+1)}|j,m-1\rangle.\]Therefore, $m$ advances in unit steps. Since the ladder must terminate at $m=\pm j$, the allowed set is
\[m=-j,-j+1,\ldots,j-1,j,\]so
\[2j\in\mathbb{Z},\]and hence
\[j=0,\frac12,1,\frac32,2,\frac52,\ldots\]with representation dimension
\[\dim(j)=2j+1.\]Half-integer angular momentum is allowed because valid representations must have integer dimension $2j+1$.
Spin $1/2$ is the smallest nontrivial fractional representation of the rotation algebra.
Representations: Orbital, Spin, and SO(3) vs SU(2)
The commutator algebra is universal, but it can act on different state spaces.
- Orbital angular momentum acts on spatial wavefunctions via differential operators.
- Spin acts on an internal Hilbert space (spinors for $1/2$, higher-dimensional spaces for $j=1,2,\ldots$).
- Total angular momentum combines orbital and spin degrees of freedom in composite systems.
For spin $1/2$, one uses Pauli matrices:
\[S_i=\frac{\hbar}{2}\sigma_i,\]with
\[\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}, \qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.\]A general spinor is
\[|\psi\rangle=a|+\rangle+b|-\rangle.\]A spinor is not an ordinary spatial vector.
But its spin expectation value transforms like a vector under rotations.
For orbital angular momentum, with $\mathbf{p}=-i\hbar\nabla$,
\[\mathbf{L}=\mathbf{r}\times\mathbf{p}=-i\hbar\,\mathbf{r}\times\nabla,\]and the eigenfunctions are spherical harmonics:
\[L^2Y_l^m(\theta,\phi)=l(l+1)\hbar^2Y_l^m(\theta,\phi),\] \[L_zY_l^m(\theta,\phi)=m\hbar Y_l^m(\theta,\phi),\]with
\[l=0,1,2,3,\ldots\]Only integer $l$ occurs for orbital motion because spatial wavefunctions are taken to be single-valued under $\phi\to\phi+2\pi$.
SO(3) is the group of ordinary 3D rotations, while SU(2) is its quantum covering group acting naturally on spinors. For spin $1/2$,
\[U(\hat n,\phi)=\exp\left(-\frac{i}{\hbar}(\mathbf{S}\cdot\hat n)\phi\right) =\exp\left(-\frac{i}{2}(\boldsymbol{\sigma}\cdot\hat n)\phi\right).\]A $360^\circ$ rotation changes a spinor by a minus sign, while a $720^\circ$ rotation returns it to itself.
SO(3) is enough for ordinary vectors and integer-spin representations.
SU(2) is needed for spinors and half-integer spin.
Addition of Angular Momenta and Basis Choice
For a system with orbital $\mathbf{L}$ and spin $\mathbf{S}$, the total angular momentum is
\[\mathbf{J}=\mathbf{L}+\mathbf{S}.\]The composite Hilbert space is
\[\mathcal{H}=\mathcal{H}_L\otimes\mathcal{H}_S.\]Two complete bases are used:
- Uncoupled basis: $|l,m_l\rangle|s,m_s\rangle$, diagonalizing $L^2,L_z,S^2,S_z$.
- Coupled basis: $|l,s;j,m\rangle$, diagonalizing $L^2,S^2,J^2,J_z$.
The allowed total $j$ values are
\[j=|l-s|,|l-s|+1,\ldots,l+s.\]The two bases are related by Clebsch–Gordan coefficients:
\[|l,s;j,m\rangle=\sum_{m_l,m_s}C_{m_lm_s}^{jm}\,|l,m_l\rangle|s,m_s\rangle.\]For example, if $l=1$ and $s=\frac12$, the uncoupled counting gives $3\times2=6$ states, while the coupled decomposition gives $j=\frac32$ (4 states) and $j=\frac12$ (2 states), again totaling 6.
The uncoupled basis describes parts separately; the coupled basis describes the whole.
Both are complete bases of the same Hilbert space.
Classical generator $\rightarrow$ quantum unitary rotation $\rightarrow$ commutator algebra $\rightarrow$ representations $\rightarrow$ orbital and spin realizations $\rightarrow$ addition rules and basis transformations
Quantum angular momentum is therefore best understood as the representation theory of rotational symmetry acting on quantum states: orbital angular momentum is the spatial representation, spin is the internal representation, and total angular momentum organizes composite rotational structure through addition and coupling.