2.2.3 Addition of Angular Momenta#

Prompts

Explain the difference between uncoupled and coupled bases. When is each one useful?
State the triangle rule for adding \(j_1\) and \(j_2\). Verify the dimension count for two spin-1/2 particles.
What are Clebsch-Gordan coefficients? Derive the singlet and triplet states for two spin-1/2 particles.
How does spin-orbit coupling \(\hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}}\) lead to fine structure? Why is the coupled basis essential?

Lecture Notes#

Overview#

When two systems each carry angular momentum — whether orbital and spin, or spin and spin — the total angular momentum \(\hat{\boldsymbol{J}} = \hat{\boldsymbol{J}}_1 + \hat{\boldsymbol{J}}_2\) obeys addition rules with no classical analog. The combined Hilbert space admits two natural bases: the uncoupled basis \(\vert j_1, m_1; j_2, m_2\rangle\) (individual projections known) and the coupled basis \(\vert J, M\rangle\) (total angular momentum known). The Clebsch-Gordan coefficients provide the unitary change of basis, and the triangle rule determines which total angular momenta are allowed.

Two Bases for the Same Space#

The combined system lives in \(\mathcal{H}_{j_1} \otimes \mathcal{H}_{j_2}\), with dimension \((2j_1+1)(2j_2+1)\).

Uncoupled basis \(\vert j_1, m_1\rangle\vert j_2, m_2\rangle\): eigenstates of \(\hat{J}_{1z}\) and \(\hat{J}_{2z}\) separately. Good quantum numbers: \(m_1, m_2\).

Coupled basis \(\vert J, M\rangle\): eigenstates of \(\hat{J}^2_{\text{tot}}\) and \(\hat{J}_{z,\text{tot}}\). Good quantum numbers: \(J, M\).

Both span the same space and are related by a unitary transformation.

Triangle Rule

When adding angular momenta \(j_1\) and \(j_2\), the allowed total angular momentum is:

(49)#\[ J \in \{\vert j_1 - j_2\vert,\; \vert j_1 - j_2\vert + 1,\; \ldots,\; j_1 + j_2\} \]

Dimension check: \(\displaystyle\sum_{J=\vert j_1-j_2\vert}^{j_1+j_2}(2J+1) = (2j_1+1)(2j_2+1)\).

Derivation: Triangle Rule

The maximum \(M\) is \(m_1 + m_2 = j_1 + j_2\), so \(J_{\max} = j_1 + j_2\). Ladder operators connect all \(M\) values within a given \(J\), so each \(J\) contributes \(2J+1\) states. Subtracting the \(J_{\max}\) multiplet and repeating gives \(J_{\min} = \vert j_1 - j_2\vert\). The dimension sum verifies the count. \(\checkmark\)

Clebsch-Gordan Coefficients#

Clebsch-Gordan Coefficients

The CG coefficients \(\langle j_1, m_1; j_2, m_2 \vert J, M\rangle\) express the coupled basis in terms of the uncoupled basis:

(50)#\[ \vert J, M\rangle = \sum_{m_1, m_2} \langle j_1, m_1; j_2, m_2 \vert J, M\rangle\;\vert j_1, m_1\rangle\vert j_2, m_2\rangle \]

Properties:

Selection rule: nonzero only if \(M = m_1 + m_2\) and the triangle rule is satisfied
Reality: all CG coefficients are real (this is a gauge choice)
Orthonormality: \(\sum_{m_1,m_2} \langle j_1, m_1; j_2, m_2 \vert J, M\rangle\langle j_1, m_1; j_2, m_2 \vert J', M'\rangle = \delta_{JJ'}\delta_{MM'}\)

In practice, CG coefficients are looked up in tables or computed by algebra systems. For simple cases they can be derived by applying ladder operators starting from the stretched state \(\vert J_{\max}, J_{\max}\rangle = \vert j_1, j_1\rangle\vert j_2, j_2\rangle\).

Note: Iterative Computation of CG Coefficients

Methodology (iterative ladder method).

Start from the stretched state, whose coefficient is fixed:

\[ \vert J_{\max},J_{\max}\rangle=\vert j_1,j_1\rangle\vert j_2,j_2\rangle, \qquad \langle j_1,j_1;j_2,j_2\vert J_{\max},J_{\max}\rangle=1. \]

Apply total lowering operator \(\hat J_- = \hat J_{1-}+\hat J_{2-}\) repeatedly to generate states with smaller \(M\) in the same multiplet.
Expand each coupled state in the uncoupled basis with \(m_1+m_2=M\), then fix unknown coefficients by normalization and orthogonality to already-known multiplets.
Repeat for lower \(J\) sectors after removing the higher-\(J\) subspace.

Why this works (brief proof idea). The coupled and uncoupled bases span the same fixed-\(M\) subspace. Since \(\hat J_-\) preserves \(J\) and lowers \(M\) by one, it gives recursion relations between neighboring \(M\) coefficients. Starting from the unique stretched state and imposing orthonormality determines all coefficients (up to an overall phase convention).

Example: \(\tfrac12\otimes\tfrac12\to 1\).

Start with

\[ \vert 1,1\rangle=\vert\uparrow\uparrow\rangle. \]

Apply \(\hat J_-\):

\[\begin{split} \begin{split} \hat J_-\vert 1,1\rangle &=(\hat J_{1-}+\hat J_{2-})\vert\uparrow\uparrow\rangle\\ &=\hbar\vert\downarrow\uparrow\rangle+\hbar\vert\uparrow\downarrow\rangle. \end{split} \end{split}\]

But also

\[ \hat J_-\vert 1,1\rangle=\hbar\sqrt2\,\vert 1,0\rangle. \]

Therefore

\[ \vert 1,0\rangle =\frac{1}{\sqrt2}\bigl(\vert\uparrow\downarrow\rangle+\vert\downarrow\uparrow\rangle\bigr). \]

Hence the two nonzero CG coefficients for \(J=1, M=0\) are

\[ \langle\tfrac12,+\tfrac12;\tfrac12,-\tfrac12\vert 1,0\rangle =\langle\tfrac12,-\tfrac12;\tfrac12,+\tfrac12\vert 1,0\rangle =\frac{1}{\sqrt2}. \]

Key Example: Two Spin-1/2 Particles#

For \(j_1 = j_2 = 1/2\): \(J \in \{0, 1\}\), giving \(1 + 3 = 4 = 2 \times 2\) states.

Triplet States (\(J = 1\), Symmetric)

\[ \vert 1, +1\rangle = \vert\uparrow\uparrow\rangle \]

\[ \vert 1, 0\rangle = \frac{1}{\sqrt{2}}(\vert\uparrow\downarrow\rangle + \vert\downarrow\uparrow\rangle) \]

(51)#\[ \vert 1, -1\rangle = \vert\downarrow\downarrow\rangle \]

All three are symmetric under particle exchange: \(\hat{P}_{12}\vert 1, M\rangle = +\vert 1, M\rangle\).

Singlet State (\(J = 0\), Antisymmetric)

(52)#\[ \vert 0, 0\rangle = \frac{1}{\sqrt{2}}(\vert\uparrow\downarrow\rangle - \vert\downarrow\uparrow\rangle) \]

The singlet is antisymmetric: \(\hat{P}_{12}\vert 0, 0\rangle = -\vert 0, 0\rangle\). It has zero total spin and is maximally entangled — measuring one particle’s spin along any axis completely determines the other’s.

Physical consequences:

Helium ground state: Two electrons in the \(1s\) orbital have a symmetric spatial wavefunction, so the spin part must be the antisymmetric singlet — the electrons are spin-paired.
Pauli exclusion: Identical fermions in the same spatial state must form a spin singlet.

Discussion

The uncoupled state \(\vert\uparrow\downarrow\rangle\) is an equal superposition of singlet and triplet (\(M = 0\)). If you measure \(\hat{J}^2_{\text{tot}}\) on this state, you get \(J = 0\) or \(J = 1\) with equal probability. Before measuring, does the system “have” a definite total spin? What does this tell us about the relationship between individual and collective quantum numbers?

Application: Spin-Orbit Coupling#

Spin-Orbit Interaction

An electron in an atom experiences a coupling between orbital and spin angular momentum:

(53)#\[ \hat{H}_{SO} = \lambda\,\hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}} \]

Using \(\hat{\boldsymbol{J}} = \hat{\boldsymbol{L}} + \hat{\boldsymbol{S}}\), we can write \(\hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}} = \frac{1}{2}(\hat{J}^2 - \hat{L}^2 - \hat{S}^2)\), which is diagonal in the coupled basis \(\vert\ell, s; j, m_j\rangle\):

\[ \langle \hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}} \rangle = \frac{\hbar^2}{2}[j(j+1) - \ell(\ell+1) - s(s+1)] \]

For an electron (\(s = 1/2\)) with orbital quantum number \(\ell \geq 1\), two values are allowed: \(j = \ell + 1/2\) and \(j = \ell - 1/2\). Their energy splitting is the fine structure — a small but measurable correction. Example: the hydrogen \(2p\) level splits into \(2p_{3/2}\) and \(2p_{1/2}\), a preview of perturbation theory in §5.1.

Summary#

Two bases: uncoupled (\(m_1, m_2\) known) vs. coupled (\(J, M\) known), related by CG coefficients.
Triangle rule: \(J \in \{\vert j_1 - j_2\vert, \ldots, j_1 + j_2\}\); dimension count: \(\sum(2J+1) = (2j_1+1)(2j_2+1)\).
Clebsch-Gordan coefficients: real, unitary, nonzero only when \(M = m_1 + m_2\) and triangle rule holds.
Two spin-1/2: triplet (\(J = 1\), symmetric, 3 states) + singlet (\(J = 0\), antisymmetric, maximally entangled).
Spin-orbit coupling: \(\hat{\boldsymbol{L}} \cdot \hat{\boldsymbol{S}}\) is diagonal in the coupled basis; causes atomic fine structure.

Homework#

1. For \(j_1 = 1\) and \(j_2 = 3/2\), list all allowed values of \(J\) using the triangle rule. Verify

\[ \sum_J (2J+1) = (2j_1+1)(2j_2+1). \]

2. For two spin-1/2 particles, derive the triplet and singlet states from scratch. Start from the stretched state \(\vert 1,1\rangle=\vert\uparrow\uparrow\rangle\), apply \(\hat J_-\) to obtain \(\vert1,0\rangle\), then determine \(\vert0,0\rangle\) by orthogonality and normalization.

3. Two identical fermions occupy the same spatial orbital \(\phi(\boldsymbol r)\). Explain why the spin state must be the singlet and why triplet spin states are forbidden.

4. Show that

\[ \hat{\boldsymbol L}\cdot\hat{\boldsymbol S}=\frac12\bigl(\hat J^2-\hat L^2-\hat S^2\bigr), \]

and use it to compute \(\langle\hat{\boldsymbol L}\cdot\hat{\boldsymbol S}\rangle\) in \(\vert \ell,\tfrac12; j,m_j\rangle\). For hydrogen \(2p\), find the fine-structure splitting between \(j=3/2\) and \(j=1/2\) in terms of \(\lambda\).

5. (CG from diagonalization) Consider spin-1 particle \(A\) and spin-1/2 particle \(B\) with

\[ \hat H=-\hat{\boldsymbol S}_A\cdot\hat{\boldsymbol S}_B. \]

Work in the uncoupled basis \(\vert1,m_A\rangle\vert\tfrac12,m_B\rangle\). Since \(\hat J_z\) commutes with \(\hat H\), first block-diagonalize by fixed \(M=m_A+m_B\). Diagonalize each \(M\) block, then identify \(J\) from the eigenvalue formula

\[ \hat{\boldsymbol S}_A\cdot\hat{\boldsymbol S}_B =\frac12\bigl(\hat J^2-\hat S_A^2-\hat S_B^2\bigr), \]

so each eigenvalue determines whether the state belongs to \(J=\tfrac32\) or \(J=\tfrac12\). Finally, check that for each fixed \(J\), the allowed \(M\) values run from \(-J\) to \(J\).

6. Build the unitary change-of-basis matrix \(U\) from uncoupled basis to coupled basis:

\[ \vert J,M\rangle=\sum_{m_A,m_B} U_{(m_A,m_B),(J,M)}\,\vert1,m_A;\tfrac12,m_B\rangle, \]

with

\[ U_{(m_A,m_B),(J,M)}=\langle1,m_A;\tfrac12,m_B\vert J,M\rangle. \]

Explain why each fixed-\(M\) sub-block of \(U\) is exactly a CG-coefficient matrix. Compute explicitly the \(M=\tfrac12\) block.

7. (Self-contained projector problem) Define projectors from coupled states:

\[ \hat P_{J}=\sum_{M=-J}^{J}\vert J,M\rangle\langle J,M\vert, \qquad J\in\left\{\tfrac12,\tfrac32\right\}. \]

Using the states from Problems 5–6, verify

\[ \hat P_{1/2}+\hat P_{3/2}=\hat I, \qquad \hat P_{1/2}\hat P_{3/2}=0. \]

Then show these projectors can be written as functions of \(\hat{\boldsymbol S}_A\cdot\hat{\boldsymbol S}_B\):

\[ \hat P_{1/2}=-\frac{2}{3}\left(\hat{\boldsymbol S}_A\cdot\hat{\boldsymbol S}_B-\frac{1}{2}\hat I\right), \qquad \hat P_{3/2}=\frac{2}{3}\left(\hat{\boldsymbol S}_A\cdot\hat{\boldsymbol S}_B+\hat I\right). \]