Chapter 13 Quiz: Spin — The Quantum Property with No Classical Analogue

Instructions: This quiz covers the core concepts from Chapter 13. 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 Stern-Gerlach experiment with silver atoms produces exactly two spots on the detector screen. This result implies that the relevant angular momentum quantum number is:

(a) $\ell = 1$, giving three spots (b) $s = 1/2$, giving two spots (c) $\ell = 0$, giving one spot (d) $s = 1$, giving three spots

Q2. The Pauli matrix $\sigma_y$ is:

(a) $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

(b) $\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$

(c) $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

(d) $\begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$

Q3. A spin-1/2 particle is in the state $|+\rangle$ (spin-up along $z$). If $\hat{S}_x$ is measured, the probabilities of obtaining $+\hbar/2$ and $-\hbar/2$ are:

(a) $P(+) = 1$, $P(-) = 0$ (b) $P(+) = 0$, $P(-) = 1$ (c) $P(+) = 1/2$, $P(-) = 1/2$ (d) $P(+) = 3/4$, $P(-) = 1/4$

Q4. The product of two different Pauli matrices satisfies:

(a) $\sigma_i \sigma_j = \delta_{ij} I$ for all $i, j$ (b) $\sigma_i \sigma_j = i\epsilon_{ijk}\sigma_k$ for $i \neq j$ (c) $\sigma_i \sigma_j = 0$ for $i \neq j$ (d) $\sigma_i \sigma_j = \sigma_k$ for $i \neq j$

Q5. On the Bloch sphere, the state $|+\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle + i|-\rangle)$ is located at:

(a) The north pole (b) The south pole (c) The $+x$ direction on the equator (d) The $+y$ direction on the equator

Q6. During Larmor precession in a magnetic field $\mathbf{B} = B_0\hat{z}$, which quantity is constant?

(a) $\langle\hat{S}_x\rangle$ (b) $\langle\hat{S}_y\rangle$ (c) $\langle\hat{S}_z\rangle$ (d) $\langle\hat{S}_x\rangle + \langle\hat{S}_y\rangle$

Q7. The Larmor frequency for an electron in a 1 T magnetic field is approximately:

(a) 42 MHz (radio frequency) (b) 28 GHz (microwave) (c) 500 THz (visible light) (d) 1 Hz

Q8. Under a rotation by $2\pi$ about any axis, a spin-1/2 spinor transforms as:

(a) $\chi \to \chi$ (unchanged) (b) $\chi \to -\chi$ (sign flip) (c) $\chi \to i\chi$ (d) $\chi \to 0$

Q9. The anomalous magnetic moment of the electron, $a_e = (g_s - 2)/2$, arises from:

(a) The electron having finite size (b) Quantum electrodynamic (QED) loop corrections (c) Special relativity alone (the Dirac equation without QED) (d) Gravitational effects on the electron

Q10. A spin-1 particle has a Hilbert space of dimension:

(a) 1 (b) 2 (c) 3 (d) 4

True/False (4 questions)

For each statement, indicate whether it is true or false and provide a brief justification.

Q11. "Spin angular momentum is caused by the physical rotation of the particle about its own axis."

Q12. "The eigenstates of $\hat{S}_z$ can simultaneously be eigenstates of $\hat{S}_x$."

Q13. "On the Bloch sphere, orthogonal quantum states correspond to antipodal points (diametrically opposite)."

Q14. "The proton's nonzero magnetic moment, despite the Dirac prediction of $g = 2$ for a point particle, was early evidence that the proton has internal structure."

Short Answer (5 questions)

Q15. A particle starts in $|+\rangle$ and passes through an SG apparatus oriented at $\theta = 60°$ from the $z$-axis (in the $xz$-plane). What is the probability of measuring spin-up along this axis? Show the calculation.

Q16. Explain what the Bloch sphere is and why it is useful for visualizing spin-1/2 states. What physical information is encoded in the polar angle $\theta$? What is encoded in the azimuthal angle $\phi$?

Q17. In the three-stage Stern-Gerlach experiment ($z$-up $\to$ $x$-up $\to$ $z$-measurement), explain step by step why the final measurement gives a 50-50 split. Include the state of the particle after each stage.

Q18. What is the spin-statistics theorem? State it and give one example of a fermion and one example of a boson, along with their spin quantum numbers.

Q19. Explain the physical significance of the Larmor frequency. How does it differ between electrons and protons, and why is this difference important for magnetic resonance imaging (MRI)?

Applied Scenarios (3 questions)

Q20. Magnetic Field Precession

An electron is prepared in the state $|\chi(0)\rangle = |+\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)$ and placed in a magnetic field $\mathbf{B} = B_0\hat{z}$ with Larmor frequency $\omega_0$.

(a) Write the state $|\chi(t)\rangle$ at time $t$.

(b) Compute $\langle\hat{S}_y\rangle(t)$.

(c) At what time is the state first equal to $|+\rangle_y$?

(d) At what time does the spin first point in the $-x$ direction?

Q21. Two Sequential Measurements

A spin-1/2 particle is in the state $|\chi\rangle = \frac{\sqrt{3}}{2}|+\rangle + \frac{1}{2}|-\rangle$.

(a) What are the Bloch sphere angles $(\theta, \phi)$ for this state?

(b) What is the probability of measuring $S_z = +\hbar/2$?

(c) After measuring $S_z$ and obtaining $+\hbar/2$, the particle is immediately measured along the $x$-axis. What is the probability of finding $S_x = +\hbar/2$?

(d) What is the total probability of the sequence (measure $S_z = +\hbar/2$, then measure $S_x = +\hbar/2$)?

Q22. Spin-1 Particle

A spin-1 particle is in the state $|1, 0\rangle$ (eigenstate of $\hat{S}_z$ with $m_s = 0$).

(a) What is $\langle\hat{S}_z\rangle$ for this state?

(b) What is $\langle\hat{S}_z^2\rangle$ for this state?

(c) Compute $\Delta S_z = \sqrt{\langle\hat{S}_z^2\rangle - \langle\hat{S}_z\rangle^2}$.

(d) If $\hat{S}_x$ is measured on this state, what are the possible outcomes and their probabilities? (Use the spin-1 $S_x$ matrix from Section 13.9.)

Answer Key

Q1: (b) — Two spots requires $2s + 1 = 2$, so $s = 1/2$.

Q2: (b)

Q3: (c) — $|+\rangle = \frac{1}{\sqrt{2}}(|+\rangle_x + |-\rangle_x)$, giving equal probabilities.

Q4: (b) — From the product rule $\sigma_i\sigma_j = \delta_{ij}I + i\epsilon_{ijk}\sigma_k$; when $i \neq j$, only the $\epsilon$ term survives.

Q5: (d) — $\theta = \pi/2$, $\phi = \pi/2$ corresponds to the $+y$ direction.

Q6: (c) — $[\hat{H}, \hat{S}_z] = 0$ since $\hat{H} \propto \hat{S}_z$.

Q7: (b) — $\omega_0 = g_s eB/(2m_e) \approx 1.76 \times 10^{11}$ rad/s $\approx 28$ GHz.

Q8: (b) — $R(2\pi) = e^{-i\pi\hat{n}\cdot\boldsymbol{\sigma}} = -I$, so $\chi \to -\chi$.

Q9: (b) — The Dirac equation gives $g_s = 2$ exactly; the deviation from 2 is a QED effect.

Q10: (c) — Dimension $= 2s + 1 = 2(1) + 1 = 3$.

Q11: False. Spin is an intrinsic quantum property with no classical analogue. The name is historical; the electron does not physically rotate.

Q12: False. $[\hat{S}_z, \hat{S}_x] = i\hbar\hat{S}_y \neq 0$, so they cannot share eigenstates.

Q13: True. The angle between antipodal Bloch vectors is $\pi$, and $\cos^2(\pi/2) = 0$, confirming zero overlap (orthogonality).

Q14: True. A point-like spin-1/2 particle should have $g \approx 2$ (Dirac prediction). The proton's $g_p = 5.586$ indicates composite structure (quarks).

Q15: $P(+) = \cos^2(\theta/2) = \cos^2(30°) = (\sqrt{3}/2)^2 = 3/4 = 0.75$.

Q16: The Bloch sphere maps every normalized spin-1/2 state to a unique point on the unit sphere via $|\chi\rangle = \cos(\theta/2)|+\rangle + e^{i\phi}\sin(\theta/2)|-\rangle$. The polar angle $\theta$ determines the relative weights of $|+\rangle$ and $|-\rangle$ (and hence measurement probabilities along $z$). The azimuthal angle $\phi$ encodes the relative phase between the two components, which determines measurement statistics along axes in the $xy$-plane.

Q17: Stage 1: SG-$z$ selects $|+\rangle$. Stage 2: $|+\rangle = \frac{1}{\sqrt{2}}|+\rangle_x + \frac{1}{\sqrt{2}}|-\rangle_x$; SG-$x$ selects $|+\rangle_x = \frac{1}{\sqrt{2}}|+\rangle + \frac{1}{\sqrt{2}}|-\rangle$. Stage 3: This state has $|\langle+|+\rangle_x|^2 = 1/2$ and $|\langle-|+\rangle_x|^2 = 1/2$, giving a 50-50 split. The $x$-measurement destroyed the $z$-information because $\hat{S}_z$ and $\hat{S}_x$ do not commute.

Q18: The spin-statistics theorem states that particles with half-integer spin (fermions) obey Fermi-Dirac statistics and the Pauli exclusion principle, while particles with integer spin (bosons) obey Bose-Einstein statistics. Example fermion: electron ($s = 1/2$). Example boson: photon ($s = 1$).

Q19: The Larmor frequency $\omega_0 = \gamma B_0$ is the rate at which a spin precesses about a magnetic field direction. For electrons, $\gamma_e \approx 1.76 \times 10^{11}$ rad/(s$\cdot$T), giving GHz frequencies. For protons, $\gamma_p \approx 2.68 \times 10^8$ rad/(s$\cdot$T), giving MHz frequencies. MRI exploits the proton Larmor frequency to image hydrogen-rich tissue; the lower frequency allows deep penetration into the body.

Q20: (a) $|\chi(t)\rangle = \frac{1}{\sqrt{2}}(e^{-i\omega_0 t/2}|+\rangle + e^{i\omega_0 t/2}|-\rangle)$. (b) $\langle\hat{S}_y\rangle = \frac{\hbar}{2}\sin(\omega_0 t)$. (c) When $\omega_0 t = \pi/2$, i.e., $t = \pi/(2\omega_0)$. (d) When $\omega_0 t = \pi$, i.e., $t = \pi/\omega_0$.

Q21: (a) $\alpha = \sqrt{3}/2 = \cos(\theta/2)$ gives $\theta = \pi/3 = 60°$; $\beta = 1/2$ is real and positive, so $\phi = 0$. (b) $P(+) = 3/4$. (c) After measuring $S_z = +\hbar/2$, the state collapses to $|+\rangle$. Then $P(S_x = +\hbar/2) = 1/2$. (d) Total: $(3/4)(1/2) = 3/8$.

Q22: (a) $\langle\hat{S}_z\rangle = 0\cdot\hbar = 0$. (b) $\langle\hat{S}_z^2\rangle = 0^2 \cdot \hbar^2 = 0$. Actually: $\hat{S}_z^2$ acting on $|1,0\rangle$ gives $0^2\hbar^2|1,0\rangle = 0$, so $\langle\hat{S}_z^2\rangle = 0$. (c) $\Delta S_z = 0$. (d) The $|1,0\rangle$ state expressed in the $S_x$ eigenbasis via the spin-1 $S_x$ matrix: the eigenvalues of $S_x$ are $+\hbar$, $0$, $-\hbar$. The middle component $|1,0\rangle$ projects onto the $S_x$ eigenstates giving $P(+\hbar) = 1/4$, $P(0) = 1/2$, $P(-\hbar) = 1/4$.