Addition of Angular Momenta

Derivation

Addition of Angular Momenta

D-220 Home PU-301 Threads symmetry · matter Depends on Angular Momentum Spectrum from Commutators, Spin-1/2 and the Pauli Matrices

Statement

Given two commuting angular momenta \( \hat{\mathbf{J}}_1 \) and \( \hat{\mathbf{J}}_2 \) with fixed magnitudes \( j_1 \) and \( j_2 \), the tensor-product space \( \mathcal{H}_{j_1} \otimes \mathcal{H}_{j_2} \), of dimension \( (2j_1+1)(2j_2+1) \), decomposes into a direct sum of irreducible eigenspaces of the total angular momentum \( \hat{\mathbf{J}} = \hat{\mathbf{J}}_1 + \hat{\mathbf{J}}_2 \) with total quantum number \( j \) running from \( |j_1-j_2| \) to \( j_1+j_2 \) in integer steps, each appearing exactly once. The change of basis from the uncoupled states \( |j_1 m_1\rangle|j_2 m_2\rangle \) to the coupled states \( |j\,m\rangle \) is effected by the Clebsch-Gordan coefficients \( \langle j_1 m_1; j_2 m_2 | j\,m\rangle \), fixed up to the Condon-Shortley phase convention.

Why it matters

Nearly every composite quantum system, from the spin-orbit coupling of a single electron to the coupling of nuclear spins, the addition of photon polarizations, and the isospin structure of hadrons, requires knowing which total-\( j \) multiplets arise when two angular momenta combine and with what amplitudes. The coupled basis diagonalizes any rotationally invariant interaction \( \hat{\mathbf{J}}_1\cdot\hat{\mathbf{J}}_2 \), so the decomposition is the first step in solving fine structure, hyperfine structure, and two-body scattering with rotational symmetry.

The Clebsch-Gordan coefficients are also the concrete realization of the Wigner-Eckart theorem: they encode all the geometry of how tensor operators connect angular-momentum multiplets, separating it cleanly from the dynamical reduced matrix element. Understanding their origin is understanding how \( SU(2) \) representations multiply.

Assumptions

The two angular momenta act on distinct tensor factors, so \( [\hat{J}_{1i},\hat{J}_{2k}]=0 \) for all components.If they did not commute, \( \hat{\mathbf{J}}_1 \) and \( \hat{\mathbf{J}}_2 \) would not be independently well-defined constants of the sub-motion, and the four operators \( \hat{\mathbf{J}}_1^2,\hat{\mathbf{J}}_2^2,\hat{\mathbf{J}}^2,\hat{J}_z \) would not form a mutually commuting set, so the coupled basis would not exist as constructed.

Each factor carries a definite irreducible representation, i.e. \( j_1 \) and \( j_2 \) are sharp.If \( \hat{\mathbf{J}}_1^2 \) or \( \hat{\mathbf{J}}_2^2 \) were not diagonal (a superposition of representations), the product space would be reducible over \( j_1,j_2 \) as well and the multiplicity of each total \( j \) could exceed one.

Each factor space is the full \( (2j+1) \)-dimensional carrier of the ladder-spectrum representation with \( \hat{J}_\pm \) acting as derived in the ladder result.If the representation were not the standard irreducible one (e.g. a non-unitary or truncated ladder), the raising and lowering operators would not close the multiplet and the recursion fixing the Clebsch-Gordan coefficients would fail to terminate correctly.

Derivation

\[ \hat{\mathbf{J}} = \hat{\mathbf{J}}_1 \otimes \mathbb{1} + \mathbb{1} \otimes \hat{\mathbf{J}}_2, \qquad [\hat{J}_i,\hat{J}_k] = i\hbar\,\epsilon_{ikl}\hat{J}_l. \]

The sum of two commuting angular momenta is itself an angular momentum: the commutator of the sums reduces to the two intra-factor commutators, since the cross terms vanish by the commuting-factors assumption. A

\[ [\hat{\mathbf{J}}^2,\hat{J}_z]=0,\quad [\hat{\mathbf{J}}^2,\hat{\mathbf{J}}_1^2]=0,\quad [\hat{\mathbf{J}}^2,\hat{\mathbf{J}}_2^2]=0,\quad [\hat{J}_z,\hat{\mathbf{J}}_{1}^2]=0. \]

The four operators \( \{\hat{\mathbf{J}}_1^2,\hat{\mathbf{J}}_2^2,\hat{\mathbf{J}}^2,\hat{J}_z\} \) mutually commute, so they possess a simultaneous eigenbasis. Note that \( \hat{\mathbf{J}}^2=\hat{\mathbf{J}}_1^2+\hat{\mathbf{J}}_2^2+2\hat{\mathbf{J}}_1\cdot\hat{\mathbf{J}}_2 \) does not commute with \( \hat{J}_{1z} \) or \( \hat{J}_{2z} \) separately. A

\[ \{ |j_1 m_1\rangle|j_2 m_2\rangle \},\qquad -j_i \le m_i \le j_i, \qquad \dim = (2j_1+1)(2j_2+1). \]

Fix \( j_1,j_2 \). The uncoupled product states are eigenstates of \( \hat{\mathbf{J}}_1^2,\hat{\mathbf{J}}_2^2,\hat{J}_{1z},\hat{J}_{2z} \) and span the product space; this is the basis we will re-couple. A

\[ \hat{J}_z\,|j_1 m_1\rangle|j_2 m_2\rangle = \hbar\,(m_1+m_2)\,|j_1 m_1\rangle|j_2 m_2\rangle \;\equiv\; \hbar\,m\,|\cdots\rangle. \]

Since \( \hat{J}_z=\hat{J}_{1z}+\hat{J}_{2z} \), each product state is already an eigenstate of \( \hat{J}_z \) with \( m=m_1+m_2 \). Thus the total-\( z \) quantum number is additive and the coupled states of given \( m \) live in the subspace of product states with \( m_1+m_2=m \). A

\[ N(m) = \#\{(m_1,m_2): m_1+m_2=m\},\qquad N(m)=N(-m). \]

Count the degeneracy of each \( \hat{J}_z \) eigenvalue. For \( j_1\ge j_2 \) the count is \( N(m)=j_1+j_2-|m|+1 \) for \( |m|\ge j_1-j_2 \), and \( N(m)=2j_2+1 \) (its maximum) for \( |m|\le j_1-j_2 \). The distribution is symmetric under \( m\to-m \). B

\[ n_j = N(m{=}j) - N(m{=}j{+}1). \]

Each multiplet of total \( j \) contributes exactly one state to every \( m \) with \( |m|\le j \) and none above. So the number of distinct multiplets with total quantum number \( j \) equals the drop in the \( m \)-degeneracy on stepping from \( m=j \) to \( m=j+1 \). Evaluating the difference of the counts in Step 5 gives \( n_j=1 \) for each \( j \) in the allowed range and \( 0 \) otherwise. C

\[ j = j_1+j_2,\; j_1+j_2-1,\; \dots,\; |j_1-j_2|,\qquad \text{each once}. \]

The largest \( m \) is \( j_1+j_2 \), occurring once, so \( j_{\max}=j_1+j_2 \). Peeling off each multiplet and repeating (Step 6) yields the Clebsch-Gordan series. The range terminates at \( |j_1-j_2| \) because below that \( N(m) \) stops growing. B

\[ \sum_{j=|j_1-j_2|}^{j_1+j_2}(2j+1) = (2j_1+1)(2j_2+1). \]

Consistency: the coupled dimensions must sum to the product dimension. The sum telescopes as an arithmetic series and reproduces the product exactly, confirming that no multiplet is missing or double-counted. C

\[ |j_1{+}j_2,\;j_1{+}j_2\rangle = |j_1\,j_1\rangle|j_2\,j_2\rangle. \]

Seed the recursion. The unique state with \( m=j_1+j_2 \) must be the top of the \( j=j_1+j_2 \) multiplet; the phase is fixed to \( +1 \) by the Condon-Shortley convention. This defines the first Clebsch-Gordan coefficient. A

\[ \hat{J}_\pm|j\,m\rangle = \hbar\sqrt{j(j{+}1)-m(m{\pm}1)}\;|j\,m{\pm}1\rangle,\quad \hat{J}_\pm=\hat{J}_{1\pm}+\hat{J}_{2\pm}. \]

Apply the total lowering operator (from the ladder-spectrum result) to both sides of the coupled state, using \( \hat{J}_-=\hat{J}_{1-}+\hat{J}_{2-} \) on the uncoupled expansion. This generates every state of the multiplet from its top state. B

\[ \sqrt{(j{+}m)(j{-}m{+}1)}\,\langle m_1 m_2|j,m{-}1\rangle = \sqrt{(j_1{-}m_1{+}1)(j_1{+}m_1)}\,\langle m_1{-}1,m_2|j,m\rangle \]
\[ +\,\sqrt{(j_2{-}m_2{+}1)(j_2{+}m_2)}\,\langle m_1,m_2{-}1|j,m\rangle. \]

Projecting the lowering identity onto \( \langle j_1 m_1;j_2 m_2| \) gives the Clebsch-Gordan recursion relation. Together with orthonormality of the coupled multiplets and the phase convention that \( \langle j_1 j_1; j_2,\,j{-}j_1|j\,j\rangle>0 \), it fixes every coefficient uniquely (all real). C

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

Assemble the general coupled state. The coefficients are the entries of a real orthogonal matrix, so the inverse transformation uses the same numbers, completing the change of basis. A

Result

\[ \mathcal{H}_{j_1}\otimes\mathcal{H}_{j_2} \;=\; \bigoplus_{j=|j_1-j_2|}^{j_1+j_2}\mathcal{H}_j, \qquad |j\,m\rangle=\!\!\sum_{m_1+m_2=m}\!\!\langle j_1 m_1;j_2 m_2|j\,m\rangle\,|j_1 m_1\rangle|j_2 m_2\rangle. \]

Reading. Two angular momenta of sharp magnitude combine into a ladder of total-spin multiplets running from the difference to the sum of the individual quantum numbers, each occurring exactly once. The Clebsch-Gordan coefficients are the fixed, real amplitudes (given Condon-Shortley phases) that write each coupled state as a superposition of uncoupled product states sharing the same total \( m=m_1+m_2 \). The dimensions balance: \( \sum_j(2j+1)=(2j_1+1)(2j_2+1) \).

Units check. The coefficients \( \langle j_1 m_1;j_2 m_2|j\,m\rangle \) are pure numbers (overlaps of normalized states), dimensionless. All quantum numbers \( j,m \) are dimensionless; the physical angular momenta they label carry \( \hbar \) via \( \hat{\mathbf{J}}^2|j m\rangle=\hbar^2 j(j+1)|j m\rangle \) and \( \hat{J}_z|j m\rangle=\hbar m|j m\rangle \), so both sides of the state expansion are consistently dimensionless kets.

Limiting cases

\( j_2=0 \): the product space is already irreducible, \( j=j_1 \) only, and the single Clebsch-Gordan coefficient equals \( 1 \). Coupling to a scalar does nothing.
\( j_1=j_2=\tfrac12 \): decomposition \( \tfrac12\otimes\tfrac12=1\oplus 0 \), the spin triplet and singlet, with the familiar \( 1/\sqrt2 \) coefficients for the \( m=0 \) states. Recovers the two-spin-half result.
Stretched states \( m=\pm(j_1+j_2) \): the coupled state is a single product state with coefficient \( 1 \); no superposition, because only one \( (m_1,m_2) \) pair reaches the extreme.
Large \( j_1,j_2 \) (semiclassical): the vector model \( \mathbf{J}=\mathbf{J}_1+\mathbf{J}_2 \) with \( |\mathbf{J}| \) ranging between \( ||\mathbf{J}_1|-|\mathbf{J}_2|| \) and \( |\mathbf{J}_1|+|\mathbf{J}_2| \) reproduces the quantum \( j \)-range as the classical triangle inequality.

Breaks when

The two angular momenta do not commute (e.g. coupling a spin to itself, or non-abelian internal degrees of freedom sharing a factor): \( \hat{\mathbf{J}}_1^2 \) and \( \hat{\mathbf{J}}_2^2 \) are no longer separately conserved and the clean \( \bigoplus_j \) with unit multiplicity fails.
Coupling three or more angular momenta: the total \( j \) generally appears with multiplicity greater than one, so \( \{\hat{\mathbf{J}}_1^2,\hat{\mathbf{J}}_2^2,\hat{\mathbf{J}}_3^2,\hat{\mathbf{J}}^2,\hat{J}_z\} \) is an incomplete commuting set and extra recoupling labels (intermediate \( j_{12} \), Wigner 6j/9j symbols) are needed to resolve states.
The relevant symmetry group is not \( SU(2) \) (e.g. \( SU(3) \) flavor, Lorentz group): tensor products of irreps need not be multiplicity-free and the simple \( |j_1-j_2|\le j\le j_1+j_2 \) rule is replaced by group-specific fusion rules.
Relativistic regimes where spin and orbital angular momentum are not separately conserved (only total \( \hat{\mathbf{J}} \) commutes with the Dirac Hamiltonian): a non-relativistic \( \hat{\mathbf{L}}\otimes\hat{\mathbf{S}} \) product decomposition is only approximate.

Failure modes

Adding the magnitudes: writing \( j=j_1+j_2 \) as the only value, or letting \( m \) run past \( j \). Only \( m=m_1+m_2 \) is additive; \( j \) takes a range.
Assuming \( \hat{J}_{1z} \) commutes with \( \hat{\mathbf{J}}^2 \). It does not, because of the cross term \( 2\hat{\mathbf{J}}_1\cdot\hat{\mathbf{J}}_2 \); this is why the coupled basis is not the uncoupled basis.
Dropping the Condon-Shortley phase and then quoting sign-sensitive coefficients (e.g. singlet vs triplet \( m=0 \)) with the wrong relative sign.
Forgetting the lower bound \( |j_1-j_2| \) and including negative or unphysical \( j \).
Treating Clebsch-Gordan coefficients as symmetric in \( 1\leftrightarrow 2 \); swapping factors introduces a phase \( (-1)^{j_1+j_2-j} \).
Using coefficients of the wrong \( m \): only product states with \( m_1+m_2=m \) contribute; including others gives a non-orthogonal mess.

Discussion

The decomposition is a statement about the representation theory of \( SU(2) \): the tensor product of two irreducible representations labelled by \( j_1 \) and \( j_2 \) reduces into a direct sum of irreducibles, and for \( SU(2) \) this reduction is multiplicity-free, which is exactly why the single quantum number \( j \) together with \( m \) fully labels the coupled states. The Clebsch-Gordan coefficients are the intertwiners realizing this isomorphism concretely in the chosen bases. Everything downstream, from selection rules to the Wigner-Eckart theorem, is a consequence of this one algebraic fact.

Physically, the operator \( \hat{\mathbf{J}}_1\cdot\hat{\mathbf{J}}_2=\tfrac12(\hat{\mathbf{J}}^2-\hat{\mathbf{J}}_1^2-\hat{\mathbf{J}}_2^2) \) is diagonal in the coupled basis with eigenvalue \( \tfrac{\hbar^2}{2}[j(j+1)-j_1(j_1+1)-j_2(j_2+1)] \). Any rotationally invariant interaction built from this dot product, spin-orbit coupling \( \xi\,\hat{\mathbf{L}}\cdot\hat{\mathbf{S}} \), the hyperfine \( A\,\hat{\mathbf{I}}\cdot\hat{\mathbf{J}} \), or the Heisenberg exchange \( J\,\hat{\mathbf{S}}_1\cdot\hat{\mathbf{S}}_2 \), is therefore solved instantly once the coupled multiplets are known. This is the practical payoff of building the coupled basis.

The coefficients carry all the geometry of how rotational multiplets combine, and this geometry is universal: the same \( \langle j_1 m_1;j_2 m_2|j m\rangle \) that couples two spins also governs how a rank-\( k \) spherical tensor operator \( \hat{T}^{(k)}_q \) connects states, via the Wigner-Eckart theorem \( \langle \alpha' j' m'|\hat{T}^{(k)}_q|\alpha j m\rangle=\langle j m; k q|j' m'\rangle\,\langle \alpha' j'\|\hat{T}^{(k)}\|\alpha j\rangle \). The dynamical content sits entirely in the reduced matrix element; the \( m \)-dependence is pure Clebsch-Gordan geometry.

A symmetric repackaging is the Wigner 3j symbol, \( \langle j_1 m_1;j_2 m_2|j_3,-m_3\rangle=(-1)^{j_1-j_2-m_3}\sqrt{2j_3+1}\begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3\end{pmatrix} \), which manifests the permutation and inversion symmetries of the coupling (invariance under even permutations of columns, and a phase \( (-1)^{j_1+j_2+j_3} \) under odd ones and under \( m_i\to-m_i \)). These symmetries and the triangle condition \( |j_1-j_2|\le j_3\le j_1+j_2 \) are, in modern language, the fusion rules of the \( SU(2) \) fusion category, and their deformation to \( q \)-deformed \( 6j \) symbols underlies topological quantum field theory and anyonic braiding. The humble triangle inequality is the classical shadow of a deep categorical structure.

Common misconceptions. The coupled and uncoupled bases are two complete descriptions of the same Hilbert space, not different physical situations; a system is not "in" one basis. Also, \( j \) is not observed to be a superposition, a state of definite \( j \) and \( m \) is a definite physical state, it is only its expansion in product states that involves several \( (m_1,m_2) \). Finally, the maximal-\( m \) (stretched) states are the only ones that look identical in both bases; every intermediate \( m \) genuinely mixes.

Worked examples

Two spin-\( \tfrac12 \) particles: \( \tfrac12\otimes\tfrac12 \). Find all coupled states.

Range: \( |{\tfrac12-\tfrac12}|\le j\le\tfrac12+\tfrac12 \Rightarrow j=0,1 \). Dimensions \( (2\cdot1+1)+(2\cdot0+1)=3+1=4=2\times2 \). Check. A

\[ |1,1\rangle=|\!\uparrow\uparrow\rangle,\qquad \hat{J}_-|1,1\rangle=\hbar\sqrt{1\cdot2-1\cdot0}\,|1,0\rangle=\hbar\sqrt2\,|1,0\rangle. \]

Seed with the stretched state (coefficient 1), then lower. On the right \( (\hat{J}_{1-}+\hat{J}_{2-})|\!\uparrow\uparrow\rangle=\hbar(|\!\downarrow\uparrow\rangle+|\!\uparrow\downarrow\rangle) \). B

\[ |1,0\rangle=\tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle+|\!\downarrow\uparrow\rangle\big),\qquad |1,-1\rangle=|\!\downarrow\downarrow\rangle. \]

Divide by \( \sqrt2 \). The singlet is the orthogonal combination with the Condon-Shortley sign (positive coefficient on \( |\!\uparrow\downarrow\rangle \)). B

\[ \underbrace{\{|1,1\rangle,|1,0\rangle,|1,-1\rangle\}}_{\text{triplet}},\qquad |0,0\rangle=\tfrac{1}{\sqrt2}\big(|\!\uparrow\downarrow\rangle-|\!\downarrow\uparrow\rangle\big). \]

Reading. The four product states reorganize into a symmetric triplet (\( j=1 \)) and an antisymmetric singlet (\( j=0 \)); the Clebsch-Gordan coefficients for the \( m=0 \) states are \( \pm1/\sqrt2 \). Units check. All coefficients dimensionless; \( (1/\sqrt2)^2+(1/\sqrt2)^2=1 \) confirms normalization.

Spin-orbit for a \( p \)-electron: couple \( \ell=1 \) and \( s=\tfrac12 \). Find the \( j \)-multiplets and the \( \hat{\mathbf{L}}\cdot\hat{\mathbf{S}} \) shifts.

Range: \( |1-\tfrac12|\le j\le 1+\tfrac12 \Rightarrow j=\tfrac12,\tfrac32 \). Dimensions \( (2+2)=6=(2\cdot1+1)(2\cdot\tfrac12+1)=3\times2 \). Check. A

\[ \hat{\mathbf{L}}\cdot\hat{\mathbf{S}}=\tfrac12\big(\hat{\mathbf{J}}^2-\hat{\mathbf{L}}^2-\hat{\mathbf{S}}^2\big),\quad \langle\hat{\mathbf{L}}\cdot\hat{\mathbf{S}}\rangle=\tfrac{\hbar^2}{2}\big[j(j{+}1)-\ell(\ell{+}1)-s(s{+}1)\big]. \]

Symbolic before numbers: diagonalize the dot product in the coupled basis. Now insert \( \ell=1,s=\tfrac12 \). B

\[ j=\tfrac32:\;\tfrac{\hbar^2}{2}\big[\tfrac{15}{4}-2-\tfrac34\big]=+\tfrac{\hbar^2}{2};\qquad j=\tfrac12:\;\tfrac{\hbar^2}{2}\big[\tfrac34-2-\tfrac34\big]=-\hbar^2. \]

Evaluate. The \( j=\tfrac32 \) level (\( p_{3/2} \)) lies above \( j=\tfrac12 \) (\( p_{1/2} \)) for a positive spin-orbit constant \( \xi>0 \). B

\[ E_{3/2}-E_{1/2}=\xi\Big(\tfrac{\hbar^2}{2}-(-\hbar^2)\Big)=\tfrac{3}{2}\,\xi\hbar^2. \]

Reading. A single \( p \)-electron splits into a \( p_{3/2} \) quartet and a \( p_{1/2} \) doublet separated by \( \tfrac32\xi\hbar^2 \); the multiplicities \( 4+2=6 \) match the product dimension. This is the origin of the fine-structure doublet (e.g. the sodium D lines). Units check. \( \xi \) has units of energy per \( \hbar^2 \), so \( \xi\hbar^2 \) is an energy; the level shifts are energies as required.

Problems

Couple \( j_1=1 \) and \( j_2=1 \). List the allowed total \( j \) and verify the dimension sum.
Solution
Range \( |1-1|\le j\le 1+1 \Rightarrow j=0,1,2 \). Dimensions \( (2\cdot0+1)+(2\cdot1+1)+(2\cdot2+1)=1+3+5=9=(2\cdot1+1)(2\cdot1+1)=3\times3 \). The \( j=2 \) multiplet is symmetric, \( j=0 \) symmetric, and \( j=1 \) antisymmetric under exchange (phase \( (-1)^{j_1+j_2-j}=(-1)^{2-j} \)).
Write the coupled state \( |j=\tfrac32,\,m=\tfrac12\rangle \) for \( \ell=1,s=\tfrac12 \) in the uncoupled basis \( |m_\ell,m_s\rangle \).
Solution
Lower \( |\tfrac32,\tfrac32\rangle=|m_\ell{=}1,m_s{=}{\uparrow}\rangle \): \( \hat{J}_-|\tfrac32,\tfrac32\rangle=\hbar\sqrt{\tfrac32\cdot\tfrac52-\tfrac32\cdot\tfrac12}\,|\tfrac32,\tfrac12\rangle=\hbar\sqrt3\,|\tfrac32,\tfrac12\rangle \). On the right, \( \hat{L}_-|1,\uparrow\rangle=\hbar\sqrt2\,|0,\uparrow\rangle \) and \( \hat{S}_-|1,\uparrow\rangle=\hbar\,|1,\downarrow\rangle \). Hence \( |\tfrac32,\tfrac12\rangle=\sqrt{\tfrac23}\,|0,\uparrow\rangle+\sqrt{\tfrac13}\,|1,\downarrow\rangle \). Coefficients: \( \sqrt{2/3},\sqrt{1/3} \); normalized since \( 2/3+1/3=1 \).
Show that \( \sum_{j=|j_1-j_2|}^{j_1+j_2}(2j+1)=(2j_1+1)(2j_2+1) \) for general \( j_1\ge j_2 \).
Solution
The sum runs over \( j=j_1-j_2 \) to \( j_1+j_2 \), a total of \( 2j_2+1 \) terms. It is an arithmetic series in \( (2j+1) \) with first term \( 2(j_1-j_2)+1 \), last term \( 2(j_1+j_2)+1 \). Sum \( =\tfrac{(2j_2+1)}{2}\big[(2(j_1-j_2)+1)+(2(j_1+j_2)+1)\big]=\tfrac{(2j_2+1)}{2}(4j_1+2)=(2j_2+1)(2j_1+1) \). QED.
For two spin-\( \tfrac12 \) particles with Heisenberg coupling \( \hat{H}=J\,\hat{\mathbf{S}}_1\cdot\hat{\mathbf{S}}_2 \), find the energy eigenvalues and degeneracies.
Solution
\( \hat{\mathbf{S}}_1\cdot\hat{\mathbf{S}}_2=\tfrac12(\hat{\mathbf{S}}^2-\hat{\mathbf{S}}_1^2-\hat{\mathbf{S}}_2^2) \). Triplet \( S=1 \): \( \tfrac{\hbar^2}{2}[2-\tfrac34-\tfrac34]=+\tfrac{\hbar^2}{4} \), energy \( +\tfrac14 J\hbar^2 \), degeneracy 3. Singlet \( S=0 \): \( \tfrac{\hbar^2}{2}[0-\tfrac34-\tfrac34]=-\tfrac{3\hbar^2}{4} \), energy \( -\tfrac34 J\hbar^2 \), degeneracy 1. Singlet-triplet gap \( =J\hbar^2 \); for \( J>0 \) (antiferromagnetic) the singlet is the ground state.
Using the Wigner-Eckart theorem, determine the ratio of matrix elements \( \langle j\,m'|\hat{J}_z|j\,m\rangle \) treating \( \hat{J}_z=\hat{T}^{(1)}_0 \) as a spherical tensor of rank 1.
Solution
\( \hat{J}_z \) is the \( q=0 \) component of the rank-1 vector \( \hat{\mathbf{J}} \). Wigner-Eckart gives \( \langle j m'|\hat{T}^{(1)}_0|j m\rangle=\langle j m;1\,0|j m'\rangle\,\langle j\|\hat{J}\|j\rangle \). The Clebsch-Gordan coefficient \( \langle j m;1\,0|j m\rangle=m/\sqrt{j(j+1)} \) is nonzero only for \( m'=m \) (since \( q=0 \) forces \( \Delta m=0 \)). Thus \( \langle j m|\hat{J}_z|j m\rangle=\hbar m \), fixing the reduced matrix element \( \langle j\|\hat{J}\|j\rangle=\hbar\sqrt{j(j+1)} \). The theorem correctly reproduces the known diagonal spectrum and enforces the selection rule \( \Delta m=0 \) for \( \hat{J}_z \).