Chapter 14 Exercises: Addition of Angular Momentum

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 14 Exercises: Addition of Angular Momentum

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core ideas. No calculations required.

A.1 Explain in your own words why the uncoupled basis $|j_1, m_1; j_2, m_2\rangle$ and the coupled basis $|J, M\rangle$ cannot simultaneously specify all four quantum numbers $m_1, m_2, J, M$. Which operators fail to commute, and why does this prevent simultaneous specification?

A.2 A student claims: "When I add angular momenta $j_1 = 2$ and $j_2 = 1$, the total angular momentum is $J = 3$ because angular momentum is a vector and $2 + 1 = 3$." What is wrong with this reasoning? What are the correct allowed values of $J$, and why can $J$ be less than $j_1 + j_2$?

A.3 The singlet state $|0, 0\rangle = \frac{1}{\sqrt{2}}(|\!\uparrow\downarrow\rangle - |\!\downarrow\uparrow\rangle)$ has total spin zero. Does this mean neither particle has any spin? Explain carefully what "total spin zero" means in terms of the individual spins, and why the individual spin measurements are still $\pm\hbar/2$.

A.4 The Wigner-Eckart theorem says that all $m$-dependence of matrix elements is contained in a CG coefficient. In physical terms, what is the geometric content of the CG coefficient, and what is the dynamical content of the reduced matrix element? Give an analogy if it helps.

A.5 Explain why the electric dipole selection rule $\Delta\ell = \pm 1$ follows from the fact that the position operator $\hat{\mathbf{r}}$ is a rank-1 tensor combined with parity considerations. Your explanation should not rely on memorizing the rule — it should derive it from first principles.

A.6 In the uncoupled basis for $\ell = 1$, $s = 1/2$, the state $|1, 1; \frac{1}{2}, -\frac{1}{2}\rangle$ has $M = 1/2$. This state appears in both the $j = 3/2$ and $j = 1/2$ coupled multiplets. How is this possible — how can one uncoupled state contribute to two different coupled states? Does this violate any principles of quantum mechanics?

A.7 The Clebsch-Gordan coefficient $\langle j_1, m_1; j_2, m_2 | J, M\rangle$ is zero unless $M = m_1 + m_2$. Give a physical argument for why this must be true, based on the properties of the $z$-component of angular momentum.

A.8 Why does L-S coupling work well for light atoms but fail for heavy atoms? What changes in heavy atoms that makes j-j coupling more appropriate? Connect your answer to the relative magnitudes of the spin-orbit and residual Coulomb interactions.

Part B: Applied Problems (⭐⭐)

These problems require direct application of the chapter's key equations.

B.1: Triangle Rule Practice

For each of the following pairs $(j_1, j_2)$, list all allowed values of $J$ and verify the dimension count $(2j_1+1)(2j_2+1) = \sum_J (2J+1)$:

(a) $j_1 = 3/2$, $j_2 = 1/2$

(b) $j_1 = 2$, $j_2 = 1$

(c) $j_1 = 3/2$, $j_2 = 3/2$

(d) $j_1 = 5/2$, $j_2 = 1$

(e) $j_1 = 2$, $j_2 = 2$

B.2: CG Coefficients by Lowering

Two particles have $j_1 = 1$ and $j_2 = 1$. The allowed values of $J$ are $2, 1, 0$.

(a) Write down the stretched state $|2, 2\rangle$ in the uncoupled basis.

(b) Apply $\hat{J}_-$ to find $|2, 1\rangle$ in the uncoupled basis.

(c) Find $|1, 1\rangle$ by requiring it to be orthogonal to $|2, 1\rangle$ in the $M = 1$ subspace. Normalize it.

(d) Continue applying $\hat{J}_-$ to find $|2, 0\rangle$.

(e) Find $|0, 0\rangle$ by requiring it to be orthogonal to both $|2, 0\rangle$ and $|1, 0\rangle$ in the $M = 0$ subspace. Verify that it is normalized.

(f) Verify your answer to (e) by computing $\hat{J}^2|0, 0\rangle$ directly and confirming it gives zero.

B.3: Spin-Orbit Coupling Energies

An electron in hydrogen has $n = 3$, $\ell = 2$ (a $d$-state).

(a) What are the allowed values of $j$?

(b) For each value of $j$, compute $\langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle$ using the identity $\hat{\mathbf{L}} \cdot \hat{\mathbf{S}} = \frac{1}{2}(\hat{J}^2 - \hat{L}^2 - \hat{S}^2)$.

(c) The spin-orbit energy shift is $\Delta E = \frac{E_n^2}{m_e c^2} \frac{1}{n\ell(\ell+1/2)(\ell+1)} \langle \hat{\mathbf{L}} \cdot \hat{\mathbf{S}} \rangle / \hbar^2$ (in appropriate units). Compute the ratio of the splittings $\Delta E(j = 5/2) / \Delta E(j = 3/2)$.

(d) Use the spectroscopic notation $n\ell_j$ to label each level. Which level lies higher in energy?

B.4: Lande g-Factor

(a) Compute the Lande g-factor $g_j$ for the hydrogen states $2p_{1/2}$ and $2p_{3/2}$.

(b) Compute $g_j$ for the $3d_{3/2}$ and $3d_{5/2}$ states.

(c) Show that for $j = \ell + 1/2$, $g_j = 1 + \frac{1}{2\ell + 1}$, and for $j = \ell - 1/2$, $g_j = 1 - \frac{1}{2\ell + 1}$.

(d) Explain physically why $g_j$ depends on $\ell$ and $j$ — what is the projection theorem telling us about the orientation of the magnetic moment relative to $\hat{\mathbf{J}}$?

B.5: The CG Selection Rule

(a) For which of the following matrix elements is the CG coefficient in the Wigner-Eckart theorem nonzero? For each, state whether it is allowed or forbidden and explain why.

$\langle 2, 1 | \hat{T}_0^{(1)} | 3, 1\rangle$
$\langle 2, 1 | \hat{T}_0^{(1)} | 5, 1\rangle$
$\langle 1, 0 | \hat{T}_1^{(2)} | 2, -1\rangle$
$\langle 3, 2 | \hat{T}_{-1}^{(1)} | 3, 3\rangle$
$\langle 0, 0 | \hat{T}_0^{(1)} | 0, 0\rangle$

(b) For each allowed matrix element, determine the CG coefficient that appears in the Wigner-Eckart theorem.

B.6: Orthogonality Verification

Using the CG coefficients from the $\frac{1}{2} \otimes \frac{1}{2}$ coupling (Section 14.5):

(a) Verify the orthogonality relation $\sum_{m_1, m_2} \langle \frac{1}{2}, m_1; \frac{1}{2}, m_2 | 1, 0\rangle \langle \frac{1}{2}, m_1; \frac{1}{2}, m_2 | 0, 0\rangle = 0$.

(b) Verify the completeness relation $\sum_{J,M} |\langle \frac{1}{2}, \frac{1}{2}; \frac{1}{2}, -\frac{1}{2} | J, M\rangle|^2 = 1$.

(c) Express the uncoupled state $|\!\uparrow\downarrow\rangle$ as a linear combination of coupled states $|J, M\rangle$. Verify that your expansion coefficients squared sum to 1.

Part C: Analysis and Proof (⭐⭐⭐)

These problems require deeper analysis, proofs, or multistep calculations.

C.1: Coupling Three Spin-1/2 Particles

Three spin-1/2 particles are coupled to total angular momentum $J$.

(a) First couple particles 1 and 2 to intermediate spin $S_{12} = 0$ or $1$. Then couple $S_{12}$ with particle 3 ($s_3 = 1/2$). List all possible final $J$ values for each $S_{12}$.

(b) How many total states are there? Verify the dimension count.

(c) The two states with $J = 1/2$ (coming from $S_{12} = 0$ and $S_{12} = 1$) span a two-dimensional subspace. Are they orthogonal to each other?

(d) Write the fully symmetric state $|J = 3/2, M = 3/2\rangle = |\!\uparrow\uparrow\uparrow\rangle$ in the uncoupled basis. Apply $\hat{J}_-$ twice to find $|3/2, 1/2\rangle$ and $|3/2, -1/2\rangle$.

C.2: Proof of the Triangle Rule

(a) Prove that the maximum value of $J$ is $j_1 + j_2$ by showing that the $M = j_1 + j_2$ subspace is one-dimensional.

(b) Prove that the minimum value of $J$ is $|j_1 - j_2|$ by counting the number of uncoupled states with $M = |j_1 - j_2|$ and comparing it to the number of coupled multiplets that contain that $M$ value.

(c) Prove by induction (or by direct summation) that $\sum_{J = |j_1 - j_2|}^{j_1 + j_2} (2J + 1) = (2j_1 + 1)(2j_2 + 1)$.

C.3: CG Coefficient Symmetries

(a) Using the definition of CG coefficients and the properties of $\hat{J}_z$, prove that $\langle j_1, m_1; j_2, m_2 | J, M\rangle = 0$ unless $M = m_1 + m_2$.

(b) Prove the symmetry relation $\langle j_1, m_1; j_2, m_2 | J, M\rangle = (-1)^{j_1 + j_2 - J}\langle j_2, m_2; j_1, m_1 | J, M\rangle$ by considering the exchange of particles 1 and 2.

(c) Use the symmetry from (b) to show that the CG coefficient $\langle j, m; j, -m | J, 0\rangle$ is zero for odd $j_1 + j_2 - J$.

C.4: The Wigner-Eckart Theorem for Scalar Operators

(a) Show that for a scalar operator $\hat{T}_0^{(0)}$ (rank 0), the Wigner-Eckart theorem reduces to $\langle j', m' | \hat{T}_0^{(0)} | j, m\rangle = \delta_{jj'}\delta_{mm'} \cdot \text{const}$. What is this "constant" in terms of the reduced matrix element?

(b) Use this result to prove that the matrix element of any scalar operator between states of different $j$ vanishes. Give a physical example.

(c) Explain why the Hamiltonian $\hat{H}$ of a rotationally invariant system is a scalar operator, and what this implies for its matrix elements in the $|j, m\rangle$ basis.

C.5: Constructing the $1 \otimes 1$ CG Table

Complete the full CG table for $j_1 = 1 \otimes j_2 = 1$, giving all nine coupled states $|J, M\rangle$ in terms of uncoupled states $|1, m_1; 1, m_2\rangle$.

(a) Build the $J = 2$ multiplet by starting from $|2, 2\rangle = |1, 1; 1, 1\rangle$ and applying $\hat{J}_-$.

(b) Build the $J = 1$ multiplet by orthogonality and $\hat{J}_-$.

(c) Build the $J = 0$ multiplet by orthogonality.

(d) Present your results in a table. Verify that the $J = 0$ state is $|0, 0\rangle = \frac{1}{\sqrt{3}}(|1, 1; 1, -1\rangle - |1, 0; 1, 0\rangle + |1, -1; 1, 1\rangle)$.

(e) Identify which coupled states are symmetric, antisymmetric, or neither under exchange of the two particles. Show that all $J = 2$ and $J = 0$ states are symmetric, while all $J = 1$ states are antisymmetric.

C.6: Reduced Matrix Elements from Known Results

The expectation value $\langle n, \ell, m | \hat{L}_z | n, \ell, m\rangle = m\hbar$ is a known result.

(a) Write $\hat{L}_z$ as the $q = 0$ component of a rank-1 irreducible tensor operator and apply the Wigner-Eckart theorem to the matrix element $\langle \ell, m | \hat{L}_0^{(1)} | \ell, m\rangle$.

(b) Use the known result for $\hat{L}_z$ and the explicit CG coefficient $\langle \ell, m; 1, 0 | \ell, m\rangle = \frac{m}{\sqrt{\ell(\ell+1)}}$ to extract the reduced matrix element $\langle \ell \| \hat{L}^{(1)} \| \ell \rangle$.

(c) Use your reduced matrix element to predict $\langle \ell, m' | \hat{L}_+ | \ell, m\rangle$ (the raising operator) and verify it agrees with the known formula $\hbar\sqrt{\ell(\ell+1) - m(m+1)}\delta_{m', m+1}$.

Part D: Computational Problems (⭐⭐⭐⭐)

These problems require using or extending the code from code/example-01-coupling.py.

D.1: CG Coefficient Verification

Using the CG coefficient calculator from the chapter code:

(a) Generate the complete CG tables for $j_1 = 3/2, j_2 = 1/2$ and verify by hand calculation for at least three coefficients.

(b) Verify the orthogonality and completeness relations numerically for $j_1 = 2, j_2 = 1$.

(c) Construct the unitary transformation matrix $U$ between uncoupled and coupled bases for $j_1 = 1, j_2 = 1$. Verify that $U^\dagger U = I$ to machine precision.

D.2: Spin-Orbit Coupling Visualization

Extend the project checkpoint code to:

(a) Calculate and plot the spin-orbit splitting pattern for the hydrogen $n = 4$ level, showing all allowed $j$ values for each $\ell$.

(b) Create a level diagram showing the uncoupled levels $|n, \ell, m_\ell, m_s\rangle$ and the coupled levels $|n, \ell, j, m_j\rangle$, connected by lines indicating which uncoupled states contribute to each coupled state (with CG coefficients as labels).

D.3: Selection Rule Checker

Write a function check_selection_rule(j1, m1, j2, m2, k, q) that determines whether a transition $|j_1, m_1\rangle \to |j_2, m_2\rangle$ is allowed for a tensor operator of rank $k$ with component $q$. Test it against the selection rules in Table 14.8.

D.4: The Lande g-Factor Landscape

(a) Write code that computes $g_j$ for all states $|\ell, s, j\rangle$ with $\ell = 0, 1, 2, 3, 4$ and $s = 1/2$.

(b) Create a table and a plot of $g_j$ versus $\ell$ for both $j = \ell + 1/2$ and $j = \ell - 1/2$.

(c) What are the limiting values of $g_j$ as $\ell \to \infty$? Explain physically.

Part E: Challenge Problems (⭐⭐⭐⭐⭐)

E.1: Three Angular Momenta and 6j Symbols

Three particles have spins $j_1 = 1$, $j_2 = 1$, $j_3 = 1$.

(a) Coupling in the order $(j_1 \oplus j_2) \oplus j_3$: first couple $j_1$ and $j_2$ to get intermediate $j_{12} = 0, 1, 2$, then couple $j_{12}$ with $j_3$. List all final states $|j_{12}, J, M\rangle$ and count the total number.

(b) Coupling in the order $j_1 \oplus (j_2 \oplus j_3)$: first couple $j_2$ and $j_3$ to get $j_{23} = 0, 1, 2$, then couple $j_1$ with $j_{23}$. List all final states $|j_{23}, J, M\rangle$.

(c) Show that both coupling schemes give the same set of allowed $J$ values and the same total number of states.

(d) The transformation between the two coupling schemes involves a Wigner 6j symbol. Write the formal expression relating $|j_{12}, J, M\rangle$ to $\sum_{j_{23}} |j_{23}, J, M\rangle \langle j_{23}, J, M | j_{12}, J, M\rangle$.

E.2: Derivation of the Projection Theorem

(a) Starting from the Wigner-Eckart theorem, derive the projection theorem:

$$\langle j, m | \hat{V}_q | j, m'\rangle = \frac{\langle j, m | \hat{\mathbf{J}} \cdot \hat{\mathbf{V}} | j, m\rangle}{j(j+1)\hbar^2} \langle j, m | \hat{J}_q | j, m'\rangle$$

for any vector operator $\hat{\mathbf{V}}$.

(b) Apply this to derive the Lande g-factor formula, starting from $\hat{\boldsymbol{\mu}} = -\mu_B(\hat{\mathbf{L}} + g_s\hat{\mathbf{S}})/\hbar$ with $g_s = 2$.

(c) Calculate $\langle j, m | \hat{\mathbf{J}} \cdot \hat{\boldsymbol{\mu}} | j, m\rangle$ explicitly and extract $g_j$.