Chapter 12 Exercises: Angular Momentum Algebra

Part A: Conceptual Questions (⭐)

These questions test your understanding of the core algebraic structure. No lengthy calculations required.

A.1 Explain in your own words why the commutation relations $[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$ are taken as the definition of angular momentum in quantum mechanics, rather than the classical formula $\hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}$. What is gained by the algebraic definition?

A.2 The eigenvalue of $\hat{J}^2$ is $j(j+1)\hbar^2$, not $j^2\hbar^2$. This means the "length" of the angular momentum vector, $\sqrt{j(j+1)}\hbar$, is always strictly greater than the maximum projection on any axis, $j\hbar$. What does this imply physically about whether the angular momentum vector can ever be perfectly aligned along a single direction?

A.3 A student claims: "If I measure $\hat{J}_z$ and get the maximum value $j\hbar$, then $\hat{J}_x = 0$ and $\hat{J}_y = 0$, because all the angular momentum is along $z$." Explain why this reasoning is incorrect. What can you say about $\langle \hat{J}_x \rangle$ and $\langle \hat{J}_y \rangle$ in the state $|j, j\rangle$?

A.4 Why does the ladder of $m$-values terminate? That is, why can't $\hat{J}_+$ keep raising $m$ indefinitely? Identify the specific mathematical constraint that forces termination.

A.5 Explain why orbital angular momentum can only take integer values of $l$, while the general angular momentum algebra allows half-integer $j$. Where does the additional constraint for orbital angular momentum come from?

A.6 Under a rotation by $2\pi$ about any axis, a spin-1/2 state picks up a minus sign: $|j, m\rangle \to -|j, m\rangle$. Is this minus sign physically observable? Can you design an experiment that would detect it? (Hint: Think about interference.)

A.7 The Casimir operator $\hat{J}^2$ is proportional to the identity matrix within each $j$-subspace. Explain why this must be the case, using the fact that $[\hat{J}^2, \hat{J}_i] = 0$ for all $i$.

A.8 Compare the ladder operators for angular momentum ($\hat{J}_\pm$) with the ladder operators for the harmonic oscillator ($\hat{a}, \hat{a}^\dagger$). In what ways are they similar? In what crucial way do they differ? (Hint: Consider whether the ladders terminate.)

Part B: Applied Problems (⭐⭐)

These problems require direct application of the chapter's formulas.

B.1: Ladder Operator Calculations

For $j = 2$, compute the following explicitly:

(a) $\hat{J}_+ |2, 1\rangle$ (express as a ket with numerical coefficient)

(b) $\hat{J}_- |2, -1\rangle$ (express as a ket with numerical coefficient)

(c) $\hat{J}_+ \hat{J}_- |2, 0\rangle$ (use the identity $\hat{J}_+ \hat{J}_- = \hat{J}^2 - \hat{J}_z^2 + \hbar\hat{J}_z$ to check)

(d) $\hat{J}_- \hat{J}_+ |2, 2\rangle$

B.2: Matrix Construction for $j = 1$

(a) Construct the $3 \times 3$ matrix representations of $\hat{J}_x$, $\hat{J}_y$, and $\hat{J}_z$ for $j = 1$.

(b) Verify by explicit matrix multiplication that $[\hat{J}_x, \hat{J}_y] = i\hbar\hat{J}_z$.

(c) Verify that $\hat{J}_x^2 + \hat{J}_y^2 + \hat{J}_z^2 = 2\hbar^2 \hat{I}_3$.

(d) Find the eigenvalues and eigenvectors of $\hat{J}_x$ for $j = 1$. (These are the states with definite angular momentum along $x$.)

B.3: The $j = 1/2$ Algebra

Using the Pauli matrices $\hat{J}_i = (\hbar/2)\sigma_i$:

(a) Verify $\sigma_x \sigma_y = i\sigma_z$ (and cyclic permutations).

(b) Show that $\sigma_i^2 = \hat{I}$ for each $i$.

(c) Compute $(\hat{\mathbf{J}} \cdot \hat{n})^2$ where $\hat{n} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$ is an arbitrary unit vector. Show the result is $(3/4)\hbar^2 \hat{I}$.

(d) Find the eigenvalues and normalized eigenvectors of $\hat{J}_x$ for $j = 1/2$.

B.4: Uncertainty Relations

(a) For the state $|j, m\rangle$, compute $\langle \hat{J}_x \rangle$, $\langle \hat{J}_y \rangle$, $\langle \hat{J}_x^2 \rangle$, and $\langle \hat{J}_y^2 \rangle$. (Hint: Express $\hat{J}_x$ and $\hat{J}_y$ in terms of $\hat{J}_\pm$.)

(b) Show that $\Delta J_x = \Delta J_y = \hbar\sqrt{[j(j+1) - m^2]/2}$ in the state $|j, m\rangle$.

(c) Verify the uncertainty relation $\Delta J_x \cdot \Delta J_y \geq (\hbar/2)|\langle \hat{J}_z \rangle|$ for the specific case $j = 1$, $m = 1$.

(d) For which state $|j, m\rangle$ (with fixed $j$) is the product $\Delta J_x \cdot \Delta J_y$ minimized? Maximized?

B.5: Rotation Matrices

(a) Compute the full Wigner $D$-matrix $D^{(1/2)}(\alpha, \beta, \gamma)$ for the rotation with Euler angles $\alpha = \pi/2$, $\beta = \pi/2$, $\gamma = 0$.

(b) Apply this rotation to the state $|\uparrow\rangle = |1/2, 1/2\rangle$. Express the result in the $\{|\uparrow\rangle, |\downarrow\rangle\}$ basis.

(c) Compute $d^{(1)}(\pi/2)$ and verify it is an orthogonal matrix.

(d) What rotation takes $|1, 1\rangle$ to $|1, 0\rangle$? (There are multiple correct answers; find one set of Euler angles.)

B.6: Normalization Verification

(a) Starting from $|1, 1\rangle$, construct $|1, 0\rangle$ and $|1, -1\rangle$ by repeated application of $\hat{J}_-$, keeping track of all normalization constants.

(b) Verify that $\langle 1, 0 | 1, 0 \rangle = 1$ using the normalization constants you found.

(c) Repeat the construction starting from $|3/2, 3/2\rangle$ down to $|3/2, -3/2\rangle$.

B.7: Expectation Values

Consider the state $|\psi\rangle = \frac{1}{\sqrt{3}}|1, 1\rangle + \frac{1}{\sqrt{3}}|1, 0\rangle + \frac{1}{\sqrt{3}}|1, -1\rangle$ (a spin-1 system).

(a) Compute $\langle \hat{J}_z \rangle$ and $\langle \hat{J}_z^2 \rangle$.

(b) Compute $\langle \hat{J}_x \rangle$. (Hint: Use the matrix representation.)

(c) If you measure $\hat{J}^2$, what result do you get? With what probability?

(d) If you measure $\hat{J}_z$, what are the possible outcomes and their probabilities?

B.8: The $j = 2$ Matrices

(a) Construct the $5 \times 5$ matrix for $\hat{J}_+$ in the $j = 2$ representation.

(b) From it, construct $\hat{J}_x$.

(c) Find the trace of $\hat{J}_x$ and $\hat{J}_x^2$. What general rule do these suggest?

B.9: Power of the Casimir

(a) For general $j$, compute $\text{Tr}(\hat{J}_z)$ by summing $m\hbar$ over $m = -j$ to $j$. Show it equals zero.

(b) Compute $\text{Tr}(\hat{J}_z^2)$ by summing $m^2 \hbar^2$. Show it equals $\frac{1}{3}j(j+1)(2j+1)\hbar^2$.

(c) Use the cyclic invariance of the trace and the fact that $\hat{J}^2 = j(j+1)\hbar^2 \hat{I}$ to deduce $\text{Tr}(\hat{J}_x^2) = \text{Tr}(\hat{J}_y^2) = \text{Tr}(\hat{J}_z^2)$.

B.10: Rotation of a Spin-1 State

A spin-1 particle is initially in the state $|1, 1\rangle$ (angular momentum maximally along $z$).

(a) Rotate the state by angle $\beta$ about the $y$-axis. Express the rotated state as a linear combination of $|1, m\rangle$.

(b) What is the probability of measuring $m = 0$ after the rotation?

(c) For what angle $\beta$ is the probability of measuring $m = -1$ equal to 1/2?

(d) Show that the sum of all three probabilities equals 1 for any $\beta$.

Part C: Analysis Problems (⭐⭐⭐)

These problems require deeper reasoning and multi-step derivations.

C.1: Deriving the Uncertainty Relation

Starting from the general uncertainty relation $\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle [\hat{A}, \hat{B}]\rangle|$:

(a) Derive the angular momentum uncertainty relation $\Delta J_x \cdot \Delta J_y \geq \frac{\hbar}{2}|\langle \hat{J}_z \rangle|$.

(b) Show that the states $|j, j\rangle$ and $|j, -j\rangle$ saturate this inequality (they are minimum-uncertainty states for $J_x$ and $J_y$).

(c) Show that $|j, 0\rangle$ does not saturate the inequality for $j \geq 1$. What does this mean physically?

C.2: Proving Hermiticity Relations

(a) Prove that $\hat{J}_+^\dagger = \hat{J}_-$ using the Hermiticity of $\hat{J}_x$ and $\hat{J}_y$.

(b) Show that $\langle j, m+1 | \hat{J}_+ | j, m \rangle = \langle j, m | \hat{J}_- | j, m+1 \rangle^*$.

(c) Use this to prove that the phase convention $c_+^{j,m} > 0$ (real and positive) is consistent with $c_-^{j,m+1} > 0$.

C.3: Alternative Proof of the Spectrum

Instead of using the ladder argument, prove that $\lambda_j = j(j+1)\hbar^2$ by the following method:

(a) Show that $\hat{J}_x^2 + \hat{J}_y^2 = \hat{J}^2 - \hat{J}_z^2$, and that $\langle j, m | (\hat{J}_x^2 + \hat{J}_y^2) | j, m \rangle \geq 0$.

(b) Conclude that $\lambda_j \geq m^2\hbar^2$ for all $m$.

(c) From the ladder property and the existence of $m_{\max}$, show that $\hat{J}_-\hat{J}_+ |j, m_{\max}\rangle = 0$ and derive $\lambda_j = m_{\max}(m_{\max}+1)\hbar^2$.

(d) Explain why this alternative proof is essentially the same as the one in the text, just organized differently.

C.4: Matrix Exponential for $j = 1$

(a) Compute $e^{-i\beta \hat{J}_y/\hbar}$ for $j = 1$ by diagonalizing $\hat{J}_y$, exponentiating the eigenvalues, and transforming back. Verify you obtain the $d^{(1)}(\beta)$ matrix from Section 12.7.

(b) Alternatively, use the Cayley-Hamilton theorem: since $\hat{J}_y$ is a $3 \times 3$ matrix, its exponential can be written as $a_0 \hat{I} + a_1 \hat{J}_y + a_2 \hat{J}_y^2$ for suitable coefficients. Find $a_0, a_1, a_2$ as functions of $\beta$ and verify the result matches.

C.5: Spherical Components

Define the spherical components of an arbitrary vector operator $\hat{V}$ as $\hat{V}_0 = \hat{V}_z$, $\hat{V}_{\pm 1} = \mp(\hat{V}_x \pm i\hat{V}_y)/\sqrt{2}$.

(a) Show that $[\hat{J}_z, \hat{V}_q] = q\hbar \hat{V}_q$ for $q = -1, 0, +1$.

(b) Show that $[\hat{J}_\pm, \hat{V}_q] = \hbar\sqrt{(1 \mp q)(1 \pm q + 1)} \hat{V}_{q \pm 1}$.

(c) These commutation relations define a spherical tensor operator of rank 1. Explain the connection to the angular momentum algebra with $j = 1$.

C.6: The $j = 0$ Representation

(a) What are the matrix representations of $\hat{J}_x$, $\hat{J}_y$, $\hat{J}_z$ for $j = 0$?

(b) What is the only state in the $j = 0$ multiplet? What happens when you apply $\hat{J}_+$ or $\hat{J}_-$ to it?

(c) Under any rotation, how does the $j = 0$ state transform? What type of physical quantity (scalar, vector, etc.) does it correspond to?

C.7: Completeness and Orthogonality

(a) For a fixed $j$, show that $\sum_{m=-j}^{j} |j, m\rangle \langle j, m| = \hat{I}_{2j+1}$ (the identity on the $j$-subspace) by verifying it on an arbitrary state.

(b) Use the orthogonality of $D$-matrix elements, $\int d\Omega \, D^{(j_1)*}_{m_1' m_1} D^{(j_2)}_{m_2' m_2} = \frac{8\pi^2}{2j_1+1}\delta_{j_1 j_2}\delta_{m_1' m_2'}\delta_{m_1 m_2}$, to show that the $D$-matrices form a complete orthogonal set of functions on the rotation group.

Part D: Computational Problems (⭐⭐ to ⭐⭐⭐)

These problems require Python implementation. Use your angular momentum module.

D.1: Verification Suite

Write a Python function that, for a given $j$:

(a) Constructs all six operators $\hat{J}_x, \hat{J}_y, \hat{J}_z, \hat{J}_+, \hat{J}_-, \hat{J}^2$.

(b) Verifies all three commutation relations to machine precision.

(c) Verifies $\hat{J}^2 = j(j+1)\hbar^2 \hat{I}$.

(d) Runs this for $j = 1/2, 1, 3/2, 2, 5/2, 3, 10$. Report the maximum numerical error.

D.2: Eigenvalue Spectrum Visualization

Write a Python script that:

(a) For $j = 0, 1/2, 1, 3/2, 2, 5/2, 3$, plots the eigenvalue spectrum of $\hat{J}_z$ (horizontal lines at $m\hbar$ for each $j$, arranged vertically by $j$).

(b) Adds a circle at each $(j, m)$ point, with the circle radius proportional to $\sqrt{j(j+1) - m^2}$ (the "transverse" angular momentum uncertainty).

(c) This produces a visual "angular momentum tower" that shows the full structure of the spectrum.

D.3: Rotation Visualization

Write a Python script that:

(a) For $j = 1$, computes the probability $|\langle 1, m' | \hat{R}(0, \beta, 0) | 1, 1 \rangle|^2$ as a function of $\beta$ for $m' = 1, 0, -1$.

(b) Plots all three probabilities on the same graph for $\beta \in [0, 2\pi]$.

(c) Verifies that the three probabilities sum to 1 at every $\beta$.

(d) Repeats for $j = 2$ with the initial state $|2, 2\rangle$, plotting five probability curves.

D.4: The $4\pi$ Periodicity

Write a Python script that:

(a) Computes $d^{(j)}(2\pi)$ for $j = 1/2, 1, 3/2, 2, 5/2, 3$.

(b) Verifies that $d^{(j)}(2\pi) = (-1)^{2j} \hat{I}$ (i.e., $+\hat{I}$ for integer $j$ and $-\hat{I}$ for half-integer $j$).

(c) Computes $d^{(j)}(4\pi)$ and verifies it equals $+\hat{I}$ for all $j$.

(d) Plots $\text{Tr}[d^{(j)}(\beta)]/(2j+1)$ as a function of $\beta$ for $j = 1/2, 1, 3/2$, showing the periodicity difference visually.

D.5: Large-$j$ Behavior

For large $j$, angular momentum approaches classical behavior.

(a) For $j = 50$, compute $\hat{J}_x$ and find its eigenvalues. Verify they are $m\hbar$ for $m = -50, \ldots, 50$.

(b) Compute the state $|j, j\rangle_x$ (the eigenstate of $\hat{J}_x$ with maximum eigenvalue $j\hbar$), expressed in the $|j, m\rangle_z$ basis. Plot $|\langle j, m | j, j \rangle_x|^2$ vs. $m$.

(c) Show that this distribution approximates a Gaussian centered at $m = 0$ with width $\sigma \approx \sqrt{j/2}$. This is the beginning of the classical limit.