Chapter 12 Quiz: Angular Momentum Algebra
Instructions: This quiz covers the core concepts from Chapter 12. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.
Multiple Choice (10 questions)
Q1. The angular momentum commutation relation $[\hat{J}_x, \hat{J}_y] = i\hbar\hat{J}_z$ encodes the physical fact that:
(a) Angular momentum is conserved in all quantum systems (b) Rotations about different axes do not commute (c) The angular momentum vector has a definite direction (d) $\hat{J}_x$ and $\hat{J}_y$ can be simultaneously measured
Q2. The Casimir operator $\hat{J}^2 = \hat{J}_x^2 + \hat{J}_y^2 + \hat{J}_z^2$ has the special property that:
(a) It commutes only with $\hat{J}_z$ (b) It commutes with all three components $\hat{J}_x$, $\hat{J}_y$, $\hat{J}_z$ (c) Its eigenvalues are $j^2\hbar^2$ (d) It is proportional to $\hat{J}_z$ in any representation
Q3. The raising operator $\hat{J}_+$ acting on the state $|j, m\rangle$ produces:
(a) A state with quantum numbers $|j+1, m\rangle$ (b) A state with quantum numbers $|j, m+1\rangle$ (up to normalization) (c) A state with quantum numbers $|j+1, m+1\rangle$ (d) A state with quantum numbers $|j, m-1\rangle$ (up to normalization)
Q4. The eigenvalue spectrum derivation shows that $j$ can take values:
(a) $j = 0, 1, 2, 3, \ldots$ (non-negative integers only) (b) $j = 1/2, 1, 3/2, 2, \ldots$ (positive half-integers and integers only) (c) $j = 0, 1/2, 1, 3/2, 2, 5/2, \ldots$ (non-negative half-integers and integers) (d) Any non-negative real number
Q5. For a system with $j = 3/2$, the dimension of the matrix representation is:
(a) 2 (b) 3 (c) 4 (d) 6
Q6. The normalization constant in $\hat{J}_+ |j, m\rangle = c |j, m+1\rangle$ is $c = \hbar\sqrt{(j-m)(j+m+1)}$. For $j = 1$, $m = 0$, this gives:
(a) $\hbar$ (b) $\hbar\sqrt{2}$ (c) $\hbar\sqrt{3}$ (d) $2\hbar$
Q7. Under a rotation by $2\pi$ about any axis, a state with angular momentum quantum number $j$ transforms as $|j, m\rangle \to$:
(a) $|j, m\rangle$ for all $j$ (b) $-|j, m\rangle$ for all $j$ (c) $(-1)^{2j}|j, m\rangle$ (d) $(-1)^m |j, m\rangle$
Q8. The Pauli matrices $\sigma_x, \sigma_y, \sigma_z$ are related to the angular momentum operators for:
(a) $j = 0$ by $\hat{J}_i = \hbar\sigma_i$ (b) $j = 1/2$ by $\hat{J}_i = (\hbar/2)\sigma_i$ (c) $j = 1$ by $\hat{J}_i = \hbar\sigma_i$ (d) Any $j$ by $\hat{J}_i = j\hbar\sigma_i$
Q9. Orbital angular momentum is restricted to integer values of $l$ because:
(a) The commutation relations forbid half-integer values for orbital motion (b) The position-space wavefunctions $Y_l^m(\theta, \phi)$ must be single-valued under $\phi \to \phi + 2\pi$ (c) The uncertainty principle prevents half-integer orbital angular momentum (d) Orbital angular momentum is always zero
Q10. The reduced rotation matrix $d^{(1/2)}_{1/2, 1/2}(\beta) = \cos(\beta/2)$. This means that if a spin-1/2 particle starts in the $|\uparrow\rangle$ state and is rotated by $\beta$ about the $y$-axis, the probability of finding it still in $|\uparrow\rangle$ is:
(a) $\cos\beta$ (b) $\cos^2\beta$ (c) $\cos(\beta/2)$ (d) $\cos^2(\beta/2)$
True/False (4 questions)
Q11. TRUE or FALSE: The operators $\hat{J}_+$ and $\hat{J}_-$ are Hermitian operators.
Justification:
Q12. TRUE or FALSE: For a given $j$, the eigenvalue of $\hat{J}^2$ depends on $m$.
Justification:
Q13. TRUE or FALSE: The commutation relation $[\hat{J}_+, \hat{J}_-] = 2\hbar\hat{J}_z$ can be derived from the fundamental commutation relations $[\hat{J}_i, \hat{J}_j] = i\hbar\epsilon_{ijk}\hat{J}_k$.
Justification:
Q14. TRUE or FALSE: A spin-3/2 particle returns to its original state after a rotation of $2\pi$.
Justification:
Short Answer (4 questions)
Q15. Explain why the angular momentum ladder has both a top rung and a bottom rung. What would go wrong physically or mathematically if the ladder extended indefinitely in either direction?
Q16. In the eigenvalue spectrum derivation, we showed that $m_{\min} = -m_{\max}$. What is the physical meaning of this symmetry? Does it hold for every angular momentum state, or only for certain special states?
Q17. Describe the key difference between the $D$-matrix element $D^{(j)}_{m'm}(\alpha, \beta, \gamma)$ and the reduced matrix element $d^{(j)}_{m'm}(\beta)$. Why is separating out the $\alpha$ and $\gamma$ dependence useful?
Q18. The matrix representations of $\hat{J}_+$ always have nonzero elements only on the superdiagonal. Explain why this must be the case from the definition of $\hat{J}_+$ and the basis ordering convention.
Applied Scenarios (2 questions)
Q19. A spin-1 particle is prepared in the state $|1, 1\rangle$ (maximum angular momentum along $z$). An experimentalist rotates the system by angle $\beta = \pi/3$ about the $y$-axis.
(a) Using the $d^{(1)}(\beta)$ matrix, express the rotated state as a superposition of $|1, 1\rangle$, $|1, 0\rangle$, and $|1, -1\rangle$.
(b) What is the probability of measuring $m = 0$ after the rotation?
(c) What is the expectation value $\langle \hat{J}_z \rangle$ in the rotated state?
(d) Verify that $\langle \hat{J}^2 \rangle$ is unchanged by the rotation.
Q20. Consider a system with $j = 3/2$. An operator $\hat{A} = \hat{J}_+^2$ is applied to the state $|3/2, -1/2\rangle$.
(a) Compute $\hat{J}_+^2 |3/2, -1/2\rangle$ step by step, applying $\hat{J}_+$ twice.
(b) Verify your answer by computing $\langle 3/2, 3/2 | \hat{J}_+^2 | 3/2, -1/2\rangle$ using the matrix representation.
(c) Is $\hat{J}_+^2$ a Hermitian operator? Justify your answer.
(d) What is $\hat{J}_+^3 |3/2, -1/2\rangle$? Why?
Answer Key Reference
Answers to odd-numbered questions are provided in Appendix H. Even-numbered answers are available in the Instructor Guide.
Quick checks for self-grading: - Q1: (b) — Rotations about different axes do not commute - Q3: (b) — $\hat{J}_+$ raises $m$ by 1, keeping $j$ fixed - Q5: (c) — Dimension is $2j+1 = 2(3/2)+1 = 4$ - Q7: (c) — The sign is $(-1)^{2j}$, distinguishing integer from half-integer - Q9: (b) — Single-valuedness of wavefunctions forces integer $l$