Angular momentum operators and SU(2) representations

Intuition [Beginner]

Classical angular momentum is $\mathbf{L}=\mathbf{r}\times \mathbf{p}$ — position crossed with momentum. It points perpendicular to the plane of rotation, and its magnitude measures the rotational motion a system carries. A planet orbiting the Sun conserves $\mathbf{L}$: direction and magnitude stay constant throughout the orbit.

In quantum mechanics, angular momentum behaves differently. The Stern-Gerlach experiment (unit 12.01.02) split a beam of silver atoms into two using a magnetic field. The angular momentum of each atom along the field axis took only the values $+\hslash /2$ or $-\hslash /2$. No continuous range appeared. Angular momentum is quantised — it comes in discrete steps.

Even more striking: electrons carry angular momentum even when they are not orbiting anything. This spin is intrinsic. You cannot stop an electron from having spin any more than you can strip it of its electric charge. Spin is angular momentum a particle carries by virtue of being what it is — no physical rotation required.

The classical formula $\mathbf{L}=\mathbf{r}\times \mathbf{p}$ cannot account for spin. A point particle at $\mathbf{r}=0$ gives $\mathbf{L}=0$ from this formula, yet electrons still carry spin $\hslash /2$. A new algebraic framework is needed.

In quantum mechanics, every observable corresponds to an operator on a Hilbert space (unit 12.02.01). Angular momentum has three component operators: $J_x$, $J_y$, $J_z$, one per spatial direction. A fourth operator $J^2=J_x^2+J_y^2+J_z^2$ measures the total angular momentum.

These operators satisfy commutation relations (unit 12.02.02):

$$[J_x,J_y]=i\hslash J_z,[J_y,J_z]=i\hslash J_x,[J_z,J_x]=i\hslash J_y.$$

The commutator $[A,B]=AB-BA$ measures the failure of two operators to share a common set of eigenstates. These three relations encode a physical constraint: you cannot simultaneously know all three components of angular momentum. Only $J^2$ and one component — conventionally $J_z$ — can have sharp values at the same time.

The simultaneous eigenstates of $J^2$ and $J_z$ are written $|j,m⟩$, labelled by two quantum numbers:

$$J^2|j,m⟩={\hslash }^{2}j(j+1)|j,m⟩,J_z|j,m⟩=\hslash m|j,m⟩.$$

Notice the eigenvalue of $J^2$ is ${\hslash }^{2}j(j+1)$, not ${\hslash }^2{j}^2$. The $j(j+1)$ factor is a consequence of the quantum algebra with no classical counterpart. This is one of the most common quantitative errors in quantum mechanics.

The quantum number $j$ takes values $0,\frac{1}{2},1,\frac{3}{2},2,\dots$ — non-negative integers and half-integers. For each $j$, the magnetic quantum number $m$ ranges from $-j$ to $+j$ in integer steps, giving $2j+1$ distinct values.

A spin-$\frac{1}{2}$ particle (electron, proton, neutron) has $j=\frac{1}{2}$, so $m=-\frac{1}{2}$ or $+\frac{1}{2}$. Two states: spin up and spin down. A spin-1 particle (the W and Z bosons) has $j=1$, so $m=-1,0,+1$. Three states.

Half-integer values of $j$ are what make spin genuinely quantum. Orbital angular momentum — the angular momentum of a particle moving through space — can only take integer $j$. A spin-$\frac{1}{2}$ particle picks up a minus sign under a full $2\pi$ rotation: $|j,m⟩\to -|j,m⟩$. It takes a $4\pi$ rotation to return to the starting state. This behaviour has no classical analogue and is what distinguishes fermions (half-integer spin) from bosons (integer spin).

The ladder operators $J_{+}=J_x+i{J}_y$ and $J_{-}=J_x-i{J}_y$ shift $m$ by one unit:

$$J_{+}|j,m⟩=\hslash \sqrt{(j-m)(j+m+1)}|j,m+1⟩$$

$$J_{-}|j,m⟩=\hslash \sqrt{(j+m)(j-m+1)}|j,m-1⟩.$$

Picture a ladder with $2j+1$ rungs. The state $|j,j⟩$ sits at the top, $|j,-j⟩$ at the bottom. The operator $J_{+}$ climbs one rung up; $J_{-}$ climbs one rung down. At the top, applying $J_{+}$ gives zero — there is nothing above. At the bottom, $J_{-}$ gives zero for the same reason.

A Stern-Gerlach magnet along $z$ splits a beam into $2j+1$ separate beams, one per value of $m$. Spin-$\frac{1}{2}$ yields two beams; spin-1 yields three. The number of beams directly measures $j$.

The commutation relations are the Lie algebra of rotations. Angular momentum in quantum mechanics generates rotational symmetry. The classification of states by $j$ and $m$ is the classification of how quantum systems transform under rotations — the representation theory of SU(2), the double cover of the rotation group SO(3).

Visual [Beginner]

For $j=\frac{1}{2}$, the ladder has two rungs. The state $|\frac{1}{2},+\frac{1}{2}⟩$ sits on top and $|\frac{1}{2},-\frac{1}{2}⟩$ on the bottom. Applying $J_{-}$ moves from top to bottom; applying $J_{+}$ moves back up.

For $j=1$, the ladder has three rungs. Each step changes $m$ by one. The ladder coefficients — the square-root factors — determine the amplitude of each step. The step from $m=0$ to $m=+1$ carries coefficient $\sqrt{2}$, reflecting the geometry of three-state angular momentum.

Worked example [Beginner]

Take a spin-1 particle ($j=1$) sent through a Stern-Gerlach apparatus along the $z$-axis. Three beams emerge, one for each value of $m$.

The three eigenstates of $J_z$ are basis vectors in a three-dimensional space:

$$|1,+1⟩=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),|1,0⟩=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),|1,-1⟩=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right).$$

Step 1. $J_z$ is diagonal in its own eigenbasis, with entries $\hslash m$:

$$J_z=\hslash \left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& -1\end{array}\right).$$

Step 2. The ladder operators follow from the ladder formulas. $J_{+}$ annihilates $|1,+1⟩$, maps $|1,0⟩$ to $\hslash \sqrt{2}|1,+1⟩$, and maps $|1,-1⟩$ to $\hslash \sqrt{2}|1,0⟩$. Reading off matrix elements:

$$J_{+}=\hslash \sqrt{2}\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right),J_{-}=\hslash \sqrt{2}\left(\begin{array}{ccc}0& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right).$$

Step 3. Recover $J_x$ and $J_y$ from $J_x=\frac{1}{2}(J_{+}+J_{-})$ and $J_y=\frac{1}{2i}(J_{+}-J_{-})$:

$$J_x=\frac{\hslash }{\sqrt{2}}\left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right),J_y=\frac{\hslash }{\sqrt{2}}\left(\begin{array}{ccc}0& -i& 0\\ i& 0& -i\\ 0& i& 0\end{array}\right).$$

Step 4. Verify the commutation relation $[J_x,J_y]=i\hslash J_z$. A direct matrix multiplication gives:

$$J_x{J}_y-J_y{J}_x=\frac{{\hslash }^2}{2}\left(\begin{array}{ccc}2i& 0& 0\\ 0& 0& 0\\ 0& 0& -2i\end{array}\right)=i\hslash \left(\begin{array}{ccc}\hslash & 0& 0\\ 0& 0& 0\\ 0& 0& -\hslash \end{array}\right)=i\hslash J_z.$$

The algebra closes. These three operators satisfy exactly the commutation relations required by the angular momentum algebra. Any other representation — spin-$\frac{1}{2}$, spin-$\frac{3}{2}$, or higher — satisfies the same relations with matrices of different size.

Check your understanding [Beginner]

Formal definition [Intermediate+]

The angular momentum algebra is defined by three Hermitian operators $J_x,J_y,J_z$ on a Hilbert space $\mathcal{H}$ satisfying

$$\boxed{[J_i,J_j]=i\hslash {ϵ}_{ijk}{J}_k,}$$

where ${ϵ}_{ijk}$ is the Levi-Civita symbol and repeated indices are summed. An operator satisfying these relations is called an angular momentum operator. The total angular momentum squared is $J^2:=J_x^2+J_y^2+J_z^2$.

Hermiticity. Each $J_i$ is required to be Hermitian ($J_i^{†}=J_i$) so that its eigenvalues are real and correspond to measurable quantities.

Commutation with $J^2$. A direct computation using the Jacobi identity and the defining commutation relations gives $[J^2,J_i]=0$ for $i=x,y,z$. This means $J^2$ commutes with every function of the angular momentum operators. In particular, $J^2$ can be diagonalised simultaneously with any one component. The conventional choice is $J_z$.

Ladder operators. Define

$$J_{+}:=J_x+i{J}_y,J_{-}:=J_x-i{J}_y.$$

These are not Hermitian; instead they are mutual adjoints: $J_{+}^{†}=J_{-}$ and $J_{-}^{†}=J_{+}$. Their commutation relations are

$$[J_z,J_{\pm }]=\pm \hslash J_{\pm },[J_{+},J_{-}]=2\hslash J_z,[J^2,J_{\pm }]=0.$$

The first relation says $J_{\pm }$ raise and lower the $J_z$ eigenvalue by $\hslash$. The third says they preserve the eigenvalue of $J^2$.

Simultaneous eigenstates. Since $[J^2,J_z]=0$, there exists an orthonormal basis of simultaneous eigenstates, denoted $|j,m⟩$:

$$J^2|j,m⟩={\hslash }^{2}j(j+1)|j,m⟩,J_z|j,m⟩=\hslash m|j,m⟩.$$

The quantum number $j$ is non-negative and $m$ takes $2j+1$ values from $-j$ to $+j$ in integer steps.

Ladder action. The ladder operators act as

$$J_{+}|j,m⟩=\hslash \sqrt{(j-m)(j+m+1)}|j,m+1⟩,$$

$$J_{-}|j,m⟩=\hslash \sqrt{(j+m)(j-m+1)}|j,m-1⟩.$$

These formulas follow from the algebra alone. The derivation uses the identities $J_{+}{J}_{-}=J^2-J_z(J_z-\hslash )$ and $J_{-}{J}_{+}=J^2-J_z(J_z+\hslash )$, combined with the requirement that all states have non-negative norm.

Resolution of the identity. For fixed $j$, the $2j+1$ states $\{|j,m⟩{\}}_{m=-j}^j$ form an orthonormal basis of the $(2j+1)$-dimensional subspace carrying the spin-$j$ representation:

$$\sum _{m=-j}^j|j,m⟩⟨j,m|=1_j.$$

Connection to SU(2). The commutation relations are the Lie algebra $\mathfrak{su}(2)$ (up to the factor $i\hslash$). The group SU(2) is the group of $2\times 2$ unitary matrices with unit determinant. Its irreducible unitary representations are in one-to-one correspondence with the allowed values of $j$. The group action is generated by exponentiation:

$$R(\hat{n},\theta )=\exp \left(-\frac{i\theta }{\hslash }\hat{n}\cdot \mathbf{J}\right),$$

which rotates the quantum state by angle $\theta$ about the axis $\hat{n}$.

Counterexamples to common slips

• $J_x,J_y,J_z$ do not commute pairwise. There is no basis diagonalising all three simultaneously. Only $J^2$ and one component share eigenstates.
• The eigenvalue ${\hslash }^{2}j(j+1)$ is not ${\hslash }^2{j}^2$. For $j=\frac{1}{2}$, $j(j+1)=\frac{3}{4}$, not $\frac{1}{4}$.
• $J_{+}$ and $J_{-}$ are not Hermitian. They are adjoints of each other: $⟨j,m^{\prime }|J_{+}|j,m⟩=⟨j,m|J_{-}|j,m^{\prime }{⟩}^{\ast }$.
• Half-integer $j$ is allowed for spin but not for orbital angular momentum. The operator $\mathbf{L}=\mathbf{r}\times \mathbf{p}$ has eigenvalues ${\hslash }^2\ell (\ell +1)$ with $\ell =0,1,2,\dots$ only, because the wavefunction must be single-valued under $2\pi$ rotation. Spin has no such constraint.

Key theorem with proof [Intermediate+]

Theorem (Classification of angular momentum representations). The irreducible representations of the angular momentum algebra $[J_i,J_j]=i\hslash {ϵ}_{ijk}{J}_k$ are in one-to-one correspondence with $j=0,\frac{1}{2},1,\frac{3}{2},\dots$. Each irreducible representation has dimension $2j+1$, spanned by $|j,m⟩$ with $m=-j,\dots ,+j$. The ladder operators act as $J_{\pm }|j,m⟩=\hslash \sqrt{(j\mp m)(j\pm m+1)}|j,m\pm 1⟩$ and annihilate the endpoints: $J_{+}|j,j⟩=J_{-}|j,-j⟩=0$.

Proof. The proof constructs each irreducible representation from the algebra alone.

Step 1: Positivity of $J^2$. Since $J^2=J_x^2+J_y^2+J_z^2$ and each $J_i$ is Hermitian, $J_i^2$ has non-negative eigenvalues. For any state $|\psi ⟩$:

$$⟨\psi |J^2|\psi ⟩=⟨J_x\psi |J_x\psi ⟩+⟨J_y\psi |J_y\psi ⟩+⟨J_z\psi |J_z\psi ⟩\ge 0.$$

Step 2: Existence of a highest-weight state. Let $|\lambda ,m⟩$ be a simultaneous eigenstate of $J^2$ and $J_z$, with $J^2|\lambda ,m⟩=\lambda |\lambda ,m⟩$ and $J_z|\lambda ,m⟩=\hslash m|\lambda ,m⟩$. From $[J_z,J_{+}]=\hslash J_{+}$, the state $J_{+}|\lambda ,m⟩$ is an eigenstate of $J_z$ with eigenvalue $\hslash (m+1)$.

Repeated application of $J_{+}$ raises $m$ without bound, but $m^2$ is bounded by $\lambda /{\hslash }^2$ since $J^2-J_z^2=J_x^2+J_y^2\ge 0$. Therefore there exists a highest-weight state $|j,j⟩$ satisfying $J_{+}|j,j⟩=0$, with $J_z|j,j⟩=\hslash j|j,j⟩$.

Step 3: Determining $\lambda$. Using $J^2=J_z^2+\hslash J_z+J_{-}{J}_{+}$ (which follows from $J_{-}{J}_{+}=J^2-J_z(J_z+\hslash )$), apply both sides to $|j,j⟩$:

$$J^2|j,j⟩=({\hslash }^2{j}^2+{\hslash }^{2}j+0)|j,j⟩={\hslash }^{2}j(j+1)|j,j⟩.$$

So $\lambda ={\hslash }^{2}j(j+1)$.

Step 4: Termination at $-j$. Applying $J_{-}$ repeatedly to $|j,j⟩$ produces eigenstates $|j,j-1⟩,|j,j-2⟩,\dots$ with decreasing $m$. By the same positivity argument, there must be a lowest-weight state $|j,m_{\min }⟩$ with $J_{-}|j,m_{\min }⟩=0$. Using $J^2=J_z^2-\hslash J_z+J_{+}{J}_{-}$:

$$J^2|j,m_{\min }⟩=({\hslash }^2{m}_{\min }^2-{\hslash }^2{m}_{\min })|j,m_{\min }⟩={\hslash }^2{m}_{\min }(m_{\min }-1)|j,m_{\min }⟩.$$

Setting this equal to ${\hslash }^{2}j(j+1)$ gives $m_{\min }(m_{\min }-1)=j(j+1)$. The solutions are $m_{\min }=j+1$ (impossible since $m_{\min }\le j$) and $m_{\min }=-j$. So the ladder terminates at $m=-j$.

Step 5: Quantisation of $j$. Starting from $m=j$ and applying $J_{-}$, $m$ decreases by 1 each step. To reach $-j$ in a finite number $n$ of steps requires $j-n=-j$, i.e., $2j=n$ for some non-negative integer $n$. Therefore $j=0,\frac{1}{2},1,\frac{3}{2},\dots$.

The dimension is $j-(-j)+1=2j+1$.

Step 6: Ladder coefficients. Compute the norm of $J_{+}|j,m⟩$:

$$\Vert J_{+}|j,m⟩{\Vert }^2=⟨j,m|J_{-}{J}_{+}|j,m⟩=⟨j,m|(J^2-J_z^2-\hslash J_z)|j,m⟩={\hslash }^2[j(j+1)-m^2-m].$$

Factorising: $j(j+1)-m(m+1)=(j-m)(j+m+1)$. Taking the positive square root (phase convention) gives $J_{+}|j,m⟩=\hslash \sqrt{(j-m)(j+m+1)}|j,m+1⟩$. The $J_{-}$ formula follows by the same argument with $J_{+}{J}_{-}=J^2-J_z(J_z-\hslash )$. $\square$

Corollary. Every finite-dimensional irreducible representation of SU(2) is equivalent to one of the spin-$j$ representations constructed above, and the spin-$j$ representation is realised by $(2j+1)\times (2j+1)$ matrices satisfying the angular momentum commutation relations.

Bridge. This classification builds toward 12.06.01 pending the hydrogen atom, where the integer-$\ell$ representations provide the angular solutions $Y_{\ell }^m$, and appears again in 12.10.01 pending quantum field theory, where the Lorentz group extends the single SU(2) to $\mathrm{SU}(2)\times \mathrm{SU}(2)$. The foundational reason angular momentum is solvable in quantum mechanics is that $\mathfrak{su}(2)$ is the simplest non-abelian Lie algebra — its representation theory closes after a single parameter $j$. This is exactly the structure that the Clebsch-Gordan machinery in the Master section exploits to decompose composite systems.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib's coverage of the mathematical substrate is substantial. The representation theory of compact Lie groups, including SU(2), is formalised: irreducible representations are classified by highest weight, and the Clebsch-Gordan decomposition of tensor products is available in Mathlib.RepresentationTheory. The Lie algebra $\mathfrak{su}(2)$ and its identification with $\mathfrak{so}(3)$ are formalised, as are the root-system and weight-lattice structures that underpin the $j$-classification.

The Casimir element of $\mathfrak{su}(2)$ is constructible in Mathlib from the Killing form. The universal enveloping algebra $U(\mathfrak{su}(2))$ and its centre are formalised in the Lie-algebra infrastructure, and the Harish-Chandra homomorphism for ${\mathfrak{sl}}_2(\mathbb{C})$ is within reach of existing machinery. The matrix groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ are formalised as subgroups of the general linear group, and the double-cover homomorphism $\mathrm{SU}(2)\to \mathrm{SO}(3)$ can be stated and proved using the existing Matrix.SpecialUnitaryGroup and Matrix.OrthogonalGroup definitions.

What Mathlib does not formalise is the physics layer: the identification of the generators $J_x,J_y,J_z$ with quantum-mechanical observables on a Hilbert space, the spectral postulate that $J^2$ and $J_z$ have the eigenvalues ${\hslash }^{2}j(j+1)$ and $\hslash m$, the ladder-operator construction as a spectral-theoretic tool, or the Wigner-Eckart theorem as a statement about matrix elements of tensor operators. The 3-j, 6-j, and 9-j recoupling symbols are absent, and the connection between the Clebsch-Gordan decomposition and physical selection rules is not formalised. Bridging this gap requires a physics-layer theory of angular momentum observables on top of Mathlib's representation theory. lean_status: none reflects this state; the lean_mathlib_gap field in the frontmatter tracks the specific missing pieces.

Advanced results [Master]

Addition of angular momenta

Consider two subsystems carrying angular momenta ${\mathbf{J}}_1$ and ${\mathbf{J}}_2$ acting on Hilbert spaces ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$. The operators from different subsystems commute: $[J_{1i},J_{2j}]=0$. Define the total angular momentum $\mathbf{J}={\mathbf{J}}_1+{\mathbf{J}}_2$. As shown in Exercise 10, $\mathbf{J}$ satisfies the same angular momentum algebra.

The product space ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$ has dimension $(2j_1+1)(2j_2+1)$. Two natural orthonormal bases span this space. The uncoupled basis $\{|j_1,m_1⟩\otimes |j_2,m_2⟩\}$ consists of eigenstates of $J_1^2,J_{1z},J_2^2,J_{2z}$. The coupled basis $\{|(j_1{j}_2)J,M⟩\}$ consists of eigenstates of $J^2,J_z,J_1^2,J_2^2$.

The Clebsch-Gordan series states that $J$ ranges from $|j_1-j_2|$ to $j_1+j_2$ in integer steps, and the dimension identity

$$\sum _{J=|j_1-j_2|}^{j_1+j_2}(2J+1)=(2j_1+1)(2j_2+1)$$

holds. This is the representation-theoretic statement $D^{(j_1)}\otimes D^{(j_2)}={⨁}_{J=|j_1-j_2|}^{j_1+j_2}{D}^{(J)}$, a special case of the general theory of tensor products of SU(2) representations.

Physical example: spin-orbit coupling in hydrogen. An electron in a hydrogen atom carries orbital angular momentum $\ell$ (integer) and spin $s=\frac{1}{2}$. The total angular momentum $j$ of the electron is the coupling of $\ell$ and $s$: by the Clebsch-Gordan series, $j=\ell +\frac{1}{2}$ or $j=\ell -\frac{1}{2}$ (for $\ell \ge 1$; for $\ell =0$ only $j=\frac{1}{2}$ is allowed). Each energy level of hydrogen with quantum numbers $n,\ell$ therefore splits into two sub-levels labelled by $j$, producing the fine-structure doublet observed spectroscopically. The $\ell =1$ (p-state) splits into $j=\frac{3}{2}$ and $j=\frac{1}{2}$; the $\ell =2$ (d-state) splits into $j=\frac{5}{2}$ and $j=\frac{3}{2}$. The degeneracy count is preserved: $(2\ell +1)\cdot 2=(2j_{+}+1)+(2j_{-}+1)$.

The coupled states $|(\ell ,\frac{1}{2})j,m_j⟩$ are linear combinations of the uncoupled states $|\ell ,m_{\ell }⟩\otimes |\frac{1}{2},m_s⟩$ with Clebsch-Gordan coefficients as weights. These coefficients determine the relative strength of transitions in atomic spectroscopy and are tabulated in standard references such as Edmonds (1957) and Varshalovich, Moskalev, and Khersonskii (1988).

Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients $⟨j_1{m}_1{j}_2{m}_2|JM⟩$ (also written $C_{m_1{m}_{2}M}^{j_1{j}_{2}J}$) relate the two bases:

$$|(j_1{j}_2)J,M⟩=\sum _{m_1,m_2}⟨j_1{m}_1{j}_2{m}_2|JM⟩|j_1{m}_1⟩\otimes |j_2{m}_2⟩.$$

These coefficients are real (by the Condon-Shortley phase convention) and non-zero only when $m_1+m_2=M$ and $|j_1-j_2|\le J\le j_1+j_2$. They satisfy orthonormality relations in both indices:

$$\sum _{m_1,m_2}⟨j_1{m}_1{j}_2{m}_2|JM⟩⟨j_1{m}_1{j}_2{m}_2|J^{\prime }{M}^{\prime }⟩={\delta }_{J{J}^{\prime }}{\delta }_{M{M}^{\prime }},$$

$$\sum _{J,M}⟨j_1{m}_1{j}_2{m}_2|JM⟩⟨j_1{m}_1^{\prime }{j}_2{m}_2^{\prime }|JM⟩={\delta }_{m_1{m}_1^{\prime }}{\delta }_{m_2{m}_2^{\prime }}.$$

The coefficients are constructed recursively by applying the lowering operator $J_{-}=J_{1-}+J_{2-}$ to the unique highest-weight state $|J,J⟩=|j_1,j_1⟩\otimes |j_2,j_2⟩$ (when $J=j_1+j_2$) and then orthogonalising against states with lower $J$. This recursive procedure is algorithmic and is implemented in standard symbolic-algebra packages.

The Casimir operator and its eigenvalues

The Casimir operator of the angular momentum algebra is $C=J^2/{\hslash }^2$. It commutes with every generator: $[C,J_i]={\hslash }^{-2}[J^2,J_i]=0$ for $i=x,y,z$. By Schur's lemma, $C$ acts as a scalar on each irreducible representation. Direct computation gives $C|j,m⟩=j(j+1)|j,m⟩$, so the eigenvalue $j(j+1)$ labels the irreducible representation uniquely.

The Casimir provides an independent proof of the completeness of the $j$-classification. If two irreps $D^{(j)}$ and $D^{(j^{\prime })}$ were equivalent, the intertwining map would commute with all $J_i$ and hence with $C$, but $C$ has distinct eigenvalues $j(j+1)\ne j^{\prime }(j^{\prime }+1)$ for $j\ne j^{\prime }$, so no non-zero intertwining map exists. The Casimir is the unique (up to scale) element of degree two in the centre of the universal enveloping algebra $U(\mathfrak{su}(2))$. For $\mathfrak{su}(2)$, the centre is the polynomial algebra $\mathbb{C}[C]$, a special case of the Harish-Chandra isomorphism for semisimple Lie algebras.

Proposition (Casimir distinguishes irreps). The spin-$j$ representation $D^{(j)}$ is the unique (up to equivalence) irreducible representation of $\mathfrak{su}(2)$ on which $C$ acts as $j(j+1)$.

Proof. The ladder construction produces a $(2j+1)$-dimensional irrep with $C=j(j+1)$. By Schur's lemma, any irrep with the same Casimir eigenvalue is equivalent, since $C-j(j+1)$ annihilates it and $C$ generates the centre of $U(\mathfrak{su}(2))$. Uniqueness follows: the direct sum ${⨁}_{j^{\prime }\ne j}{D}^{(j^{\prime })}$ has $C$-eigenvalue $j^{\prime }(j^{\prime }+1)\ne j(j+1)$ on every summand, so no isomorphic copy of $D^{(j)}$ appears in the complement. $\square$

Tensor operators

An irreducible tensor operator of rank $k$ is a set of $2k+1$ operators $T_q^{(k)}$ ($q=-k,\dots ,+k$) that transform under rotations in the same way as the angular momentum eigenstates $|k,q⟩$. The defining property is expressed by the commutation relations:

$$[J_z,T_q^{(k)}]=\hslash q{T}_q^{(k)},$$

$$[J_{\pm },T_q^{(k)}]=\hslash \sqrt{(k\mp q)(k\pm q+1)}{T}_{q\pm 1}^{(k)}.$$

Equivalently, under a rotation $R$, the transformed operator satisfies $R{T}_q^{(k)}{R}^{-1}={\sum }_{q^{\prime }}{D}_{q^{\prime }q}^{(k)}(R)T_{q^{\prime }}^{(k)}$, where $D^{(k)}$ is the Wigner $D$-matrix for spin $k$.

Tensor operators generalise the concept of vector operators (rank-1 tensors). The angular momentum components $J_x,J_y,J_z$ themselves form a rank-1 tensor operator with $T_0^{(1)}=J_z$, $T_{\pm 1}^{(1)}=\mp (J_x\pm i{J}_y)/\sqrt{2}$ (the spherical basis). The position operator $\mathbf{r}$ and the electric dipole moment are also rank-1 tensor operators.

The product of two tensor operators $T_{q_1}^{(k_1)}$ and $U_{q_2}^{(k_2)}$ combines by the same Clebsch-Gordan machinery as angular momentum states:

$$[T^{(k_1)}\otimes U^{(k_2)}{]}_Q^{(K)}=\sum _{q_1,q_2}⟨k_1{q}_1{k}_2{q}_2|KQ⟩T_{q_1}^{(k_1)}{U}_{q_2}^{(k_2)}.$$

This tensor-product coupling rule is what makes the machinery of angular momentum algebra self-contained: the same Clebsch-Gordan coefficients that decompose state spaces also decompose the space of operators. A rank-0 tensor operator (scalar) commutes with all $J_i$; the Hamiltonian of a rotationally invariant system is a scalar operator. The electric quadrupole moment is a rank-2 tensor operator, and its selection rules ($\Delta j=0,\pm 1,\pm 2$, with $j=0\nrightarrow j^{\prime }=0$ and $j=\frac{1}{2}\nrightarrow j^{\prime }=\frac{1}{2}$) follow from the Wigner-Eckart theorem applied at $k=2$.

The Wigner-Eckart theorem

The Wigner-Eckart theorem is the central simplification for matrix elements of tensor operators in the angular momentum basis. It states:

$$⟨j^{\prime }{m}^{\prime }|T_q^{(k)}|jm⟩=⟨j^{\prime }\Vert T^{(k)}\Vert j⟩\cdot \frac{⟨jmkq|j^{\prime }{m}^{\prime }⟩}{\sqrt{2j^{\prime }+1}}.$$

Here $⟨j^{\prime }\Vert T^{(k)}\Vert j⟩$ is the reduced matrix element, independent of $m$, $m^{\prime }$, and $q$. The entire dependence on the magnetic quantum numbers is carried by the Clebsch-Gordan coefficient.

The physical content is a selection rule: the matrix element vanishes unless $m^{\prime }=m+q$ and $|j-k|\le j^{\prime }\le j+k$. These constraints follow directly from the triangle condition and projection condition on the Clebsch-Gordan coefficient. For an electric dipole transition ($k=1$), the selection rules are $\Delta j=0,\pm 1$ (with $j=0\nrightarrow j^{\prime }=0$) and $\Delta m=0,\pm 1$. These are the selection rules observed in atomic spectroscopy.

The Wigner-Eckart theorem reduces the computation of $(2j^{\prime }+1)(2j+1)(2k+1)$ matrix elements to a single reduced matrix element times known geometric factors. In multi-electron atoms with complex term structures, this reduction is what makes spectroscopic calculations tractable.

The theorem is proven by noting that $T_q^{(k)}|j,m⟩$ transforms under rotations like the product $|k,q⟩\otimes |j,m⟩$, which decomposes by the Clebsch-Gordan series into states $|J,M⟩$ with $|j-k|\le J\le j+k$. Projecting onto $⟨j^{\prime },m^{\prime }|$ and using Schur's lemma (the only invariant pairing between irreps is proportional to the Clebsch-Gordan coefficient) gives the result.

SU(2) and SO(3): the double-cover homomorphism

The group SU(2) is the simply connected double cover of SO(3). The covering homomorphism $\varphi :\mathrm{SU}(2)\to \mathrm{SO}(3)$ sends a $2\times 2$ unitary matrix $U$ with $\det U=1$ to a $3\times 3$ rotation matrix via the adjoint action on the Lie algebra. Explicitly, if $U=e^{-i\theta \hat{n}\cdot \mathbf{\sigma }/2}$ (where $\mathbf{\sigma }$ are the Pauli matrices), then $\varphi (U)$ is the rotation by angle $\theta$ about axis $\hat{n}$:

$$\varphi (U{)}_{ij}=\frac{1}{2}\mathrm{tr}({\sigma }_{i}U{\sigma }_j{U}^{†}).$$

The kernel of $\varphi$ is $\{+I,-I\}$. Both $+I$ and $-I$ in SU(2) map to the identity in SO(3). A $2\pi$ physical rotation corresponds to $U=-I$ in SU(2); a $4\pi$ rotation returns to $U=+I$.

Proposition (Integer vs half-integer descent). The spin-$j$ representation of SU(2) descends to a representation of SO(3) if and only if $j$ is an integer.

Proof. The kernel $\{+I,-I\}$ must act as the identity on the representation space. In the spin-$j$ representation, $-I$ acts as $e^{-i\pi (2j)}=(-1{)}^{2j}$. This equals $+1$ when $j$ is an integer and $-1$ when $j$ is a half-integer. Half-integer representations do not descend: a $2\pi$ rotation acts as $-1$, not $+1$. $\square$

The physical consequence is that fermions require SU(2) rather than SO(3) as their rotation group. The sign change under $2\pi$ rotation was observed directly in neutron interferometry by Rauch, Treimer, and Bonse (1974): a neutron beam split and recombined with one path rotated by $2\pi$ produced destructive interference, confirming the $4\pi$-periodicity of the fermion wavefunction.

Orbital angular momentum, derived from the wavefunction on $S^2$, must be single-valued under $2\pi$ rotation and is restricted to integer $\ell$. Spin angular momentum accesses the full SU(2) representation theory including half-integer $j$.

Recoupling: 3-j, 6-j, and 9-j symbols

When three or more angular momenta are coupled, the order of coupling matters. Three angular momenta $j_1,j_2,j_3$ can be coupled as $(j_1{j}_2)J_{12}$ then $(J_{12}{j}_3)J$, or as $(j_2{j}_3)J_{23}$ then $(j_1{J}_{23})J$. The two schemes produce different basis states related by a unitary transformation whose coefficients are the 6-j symbols of Racah:

$$⟨(j_1{j}_2)J_{12},j_3;J|j_1,(j_2{j}_3)J_{23};J⟩=(-1{)}^{j_1+j_2+j_3+J}\sqrt{(2J_{12}+1)(2J_{23}+1)}\left\{\begin{array}{ccc}{j}_1& j_2& J_{12}\\ j_3& J& J_{23}\end{array}\right\}.$$

The 6-j symbol is invariant under any permutation of its columns and under exchange of the upper and lower arguments in any two columns. These symmetries reduce the number of independent symbols that must be tabulated.

The 3-j symbol is a symmetrised version of the Clebsch-Gordan coefficient introduced by Wigner:

$$\left(\begin{array}{ccc}{j}_1& j_2& j_3\\ m_1& m_2& m_3\end{array}\right)=\frac{(-1{)}^{j_1-j_2-m_3}}{\sqrt{2j_3+1}}⟨j_1{m}_1{j}_2{m}_2|j_3-m_3⟩.$$

The 3-j symbol has higher permutation symmetry than the Clebsch-Gordan coefficient: it is invariant under even permutations of columns and acquires a sign $(-1{)}^{j_1+j_2+j_3}$ under odd permutations. For four angular momenta, the 9-j symbol relates two pairwise coupling schemes and decomposes as a sum over 6-j symbols.

Racah's 1942 paper established the algebraic recursion relations for these coefficients, replacing case-by-case angular momentum algebra with tabulated, highly symmetric expressions. In nuclear shell-model calculations and atomic spectroscopy, these recoupling coefficients are the basic computational primitives.

Synthesis. The representation theory of SU(2) is the foundational reason that angular momentum in quantum mechanics is completely solvable: the Casimir identifies each irrep, the ladder operators construct it, the Clebsch-Gordan machinery decomposes tensor products, and the Wigner-Eckart theorem extracts all measurable matrix elements from a single reduced matrix element. The central insight is that the angular momentum algebra $\mathfrak{su}(2)$ is the simplest non-abelian Lie algebra, and its representation theory is both complete and closed — the bridge is between the abstract highest-weight classification and the concrete spectroscopic data of atomic and nuclear physics. Putting these together, the full apparatus — from the $j(j+1)$ eigenvalue to the 9-j recoupling symbols — identifies the quantum-mechanical rotation group with the mathematical structure of SU(2) representations. This is exactly the pattern that generalises when $\mathfrak{su}(2)$ is embedded in larger Lie algebras: the highest-weight classification builds toward 07.01.01 representation theory of SU(2), while the tensor-product decomposition appears again in 12.10.01 pending quantum field theory where the Lorentz group extends the single SU(2) to $\mathrm{SU}(2)\times \mathrm{SU}(2)$.

Full proof set [Master]

Proposition (Clebsch-Gordan triangle rule). In the coupling of angular momenta $j_1$ and $j_2$, the total angular momentum $J$ takes values $J=j_1+j_2,j_1+j_2-1,\dots ,|j_1-j_2|$, each occurring once.

Proof. The product space ${\mathcal{H}}_{j_1}\otimes {\mathcal{H}}_{j_2}$ has dimension $(2j_1+1)(2j_2+1)$. The highest value of $M=m_1+m_2$ is $j_1+j_2$, achieved uniquely by $|j_1,j_1⟩\otimes |j_2,j_2⟩$. This state is annihilated by $J_{+}=J_{1+}\otimes 1+1\otimes J_{2+}$, so it is the highest-weight state of a spin-$(j_1+j_2)$ representation, contributing $2(j_1+j_2)+1$ states.

Removing this subspace, the highest remaining value of $M$ is $j_1+j_2-1$, which occurs twice: from $(m_1,m_2)=(j_1-1,j_2)$ and $(j_1,j_2-1)$. One linear combination belongs to $J=j_1+j_2$ (already removed); the orthogonal combination initiates a $J=j_1+j_2-1$ representation contributing $2(j_1+j_2-1)+1$ states.

Proceeding by induction: at each step, the multiplicity of the highest remaining $M$-value exceeds by one the number of representations already placed. The new representation accounts for this excess. The process terminates at $J=|j_1-j_2|$. The dimension identity confirms completeness:

$$\sum _{J=|j_1-j_2|}^{j_1+j_2}(2J+1)=(2j_1+1)(2j_2+1).$$

Each allowed value of $J$ occurs exactly once. $\square$

Proposition (Selection rules from the Wigner-Eckart theorem). The matrix element $⟨j^{\prime }{m}^{\prime }|T_q^{(k)}|jm⟩$ vanishes unless $m^{\prime }=m+q$ and $|j-k|\le j^{\prime }\le j+k$. For electric dipole transitions ($k=1$): $\Delta m=0,\pm 1$ and $\Delta j=0,\pm 1$ with $j=0\nrightarrow j^{\prime }=0$.

Proof. By the Wigner-Eckart theorem, the matrix element is proportional to $⟨jmkq|j^{\prime }{m}^{\prime }⟩$. This Clebsch-Gordan coefficient vanishes unless the projection condition $m+q=m^{\prime }$ and the triangle condition $|j-k|\le j^{\prime }\le j+k$ both hold. These are the selection rules for non-vanishing Clebsch-Gordan coefficients.

For $k=1$: the triangle condition gives $j^{\prime }=j-1,j,j+1$ (excluding $j^{\prime }<0$), so $\Delta j\in \{0,\pm 1\}$. The projection condition with $q=0,\pm 1$ gives $\Delta m=q\in \{0,\pm 1\}$.

The exclusion $j=0\nrightarrow j^{\prime }=0$ follows because the triangle $|0-1|\le 0\le 0+1$ requires $1\le 0$, which fails. No Clebsch-Gordan coefficient with $j_1=0,j_2=1,J=0$ exists. $\square$

Connections [Master]

• Representation theory of SU(2) 07.01.01 (pending) develops the same irreducible representations from the pure-mathematics side: highest-weight theory, characters, and the Peter-Weyl theorem. This unit is the physics manifestation of that theory, with the additional requirement that the generators act as Hermitian observables on a Hilbert space.

• Hydrogen atom 12.06.01 pending (pending) separates the Schrodinger equation into radial and angular parts. The angular part is solved entirely by the $j=\ell$ representations constructed here: the spherical harmonics $Y_{\ell }^m(\theta ,\varphi )$ are the coordinate representation of the states $|\ell ,m⟩$ for integer $\ell$. Every eigenstate of the hydrogen atom carries an angular momentum quantum number inherited from this classification.

• NMR spectroscopy 14.12.01 (pending) measures transitions between nuclear spin states. The nuclear spin angular momentum satisfies the same SU(2) algebra, and the selection rules for radio-frequency transitions follow from the Wigner-Eckart theorem applied to the magnetic dipole operator. Larmor precession, the fundamental dynamical process in NMR, is the time evolution generated by $J_z$ in a magnetic field.

• Symmetry and group theory in chemistry 16.02.01 (pending) applies representation theory to molecular symmetry groups (point groups, space groups). The machinery of character tables, irreducible representations, and direct-product decompositions is the same as the Clebsch-Gordan theory developed here, applied to finite subgroups of SO(3) rather than to SO(3) itself.

• Operators, Hermiticity, and commutators 12.02.02 pending provides the operator-algebraic prerequisites. The angular momentum algebra $[J_i,J_j]=i\hslash {ϵ}_{ijk}{J}_k$ is the first physically important example of a non-abelian operator algebra.

• Quantum field theory 12.10.01 pending (pending) extends angular momentum to relativistic systems, where the rotation group SO(3) is replaced by the Lorentz group SO(3,1) and the representations are classified by $(j_1,j_2)$ labelling the two SU(2) factors in $\text{Spin}(3,1)\cong \text{SU}(2)\times \text{SU}(2)$.

Historical and philosophical context [Master]

The discovery of spin unfolded in stages. In 1922, Stern and Gerlach observed spatial quantisation of silver atoms in an inhomogeneous magnetic field, producing two distinct beams where classical physics predicted a continuous spread. In 1925, Uhlenbeck and Goudsmit proposed that the electron carries an intrinsic angular momentum of $\hslash /2$, explaining the anomalous Zeeman effect and the fine-structure doubling of atomic spectral lines. Pauli (1925) introduced the two-valued internal degree of freedom formally, though he resisted interpreting it as physical rotation. Dirac's relativistic electron equation (1928) showed that spin $\hslash /2$ emerges automatically from the marriage of quantum mechanics and special relativity — it is not an additional postulate but a consequence of Lorentz invariance.

The algebraic framework was unified by Wigner in 1931. His monograph Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren demonstrated that the classification of quantum states by angular momentum is the representation theory of the rotation group. Wigner showed that irreducible unitary representations of SU(2) are labelled by $j=0,\frac{1}{2},1,\dots$ and have dimension $2j+1$ — exactly the quantum angular momentum multiplets. This was the first major application of group representation theory to physics, and it established the pattern that every symmetry of a physical system classifies its states.

Racah (1942) extended the algebraic machinery with his theory of tensor operators and the recoupling coefficients now called 3-j, 6-j, and 9-j symbols. These tools made the computation of matrix elements in many-electron atoms systematic, replacing case-by-case angular momentum algebra with tabulated coefficients. Edmonds's Angular Momentum in Quantum Mechanics (1957) consolidated the practical formalism. The topological meaning of the SU(2) double cover was clarified by the Bott periodicity theorem (1956) and the development of fibre-bundle methods in physics: the rotation group SO(3) has fundamental group $\mathbb{Z}/2$, and SU(2) is its universal cover. The physical distinction between fermions and bosons is thus a topological distinction — a consequence of the global, not local, structure of the rotation group.

The experimental confirmation of the double-cover structure came with neutron interferometry. Rauch, Treimer, and Bonse (1974) split a neutron beam and rotated one path by $2\pi$ using a magnetic field. The recombined beam showed destructive interference, confirming that the neutron wavefunction acquires a factor of $-1$ under a $2\pi$ rotation — direct evidence that spin-$\frac{1}{2}$ particles transform under SU(2), not SO(3).

The philosophical significance of half-integer spin is profound. A spin-$\frac{1}{2}$ particle returns to itself only after a $4\pi$ rotation, not $2\pi$. This has no classical analogue. The spin-statistics theorem (Pauli 1940) connects this geometric property to the exclusion principle: particles with half-integer spin are fermions obeying Fermi-Dirac statistics, while particles with integer spin are bosons obeying Bose-Einstein statistics. The entire structure of matter — the periodic table, the stability of atoms, the existence of neutron stars versus white dwarfs — traces back to the representation theory of SU(2).

Bibliography [Master]

• Wigner, E. P. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Academic Press, 1959 (trans. from the 1931 German edition). Ch. 15-19.

• Pauli, W. "Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren." Zeitschrift fur Physik 31 (1925), 765-783.

• Dirac, P. A. M. The Principles of Quantum Mechanics, 4th ed. Oxford University Press, 1958. Ch. IV.

• Racah, G. "Theory of Complex Spectra II." Physical Review 62 (1942), 438-462.

• Edmonds, A. R. Angular Momentum in Quantum Mechanics. Princeton University Press, 1957.

• Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 2nd ed. Cambridge University Press, 2017. Ch. 3.

• Griffiths, D. J. and Schroeter, D. F. Introduction to Quantum Mechanics, 3rd ed. Cambridge University Press, 2018. Ch. 4.

• Varshalovich, D. A., Moskalev, A. N. and Khersonskii, V. K. Quantum Theory of Angular Momentum. World Scientific, 1988.

• Biedenharn, L. C. and Louck, J. D. Angular Momentum in Quantum Physics. Addison-Wesley, 1981.

• Hall, B. C. Quantum Theory for Mathematicians. Springer GTM 267, 2013. Ch. 17.

• Weinberg, S. The Quantum Theory of Fields, Vol. 1. Cambridge University Press, 1995. Ch. 3.

• Stern, O. and Gerlach, W. "Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld." Zeitschrift fur Physik 9 (1922), 349-352.

• Uhlenbeck, G. E. and Goudsmit, S. "Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung bezuglich des inneren Verhaltens jedes einzelnen Elektrons." Naturwissenschaften 13 (1925), 953-954.

• Tong, D. Quantum Field Theory (DAMTP Cambridge lecture notes), §1: Angular momentum, spin, representations of SU(2).

• Condon, E. U. and Shortley, G. H. The Theory of Atomic Spectra. Cambridge University Press, 1935.

• Pauli, W. "The Connection between Spin and Statistics." Physical Review 58 (1940), 716-722.