Chapter 13 Exercises: Spin — The Quantum Property with No Classical Analogue
Part A: Conceptual Questions (one star)
These questions test your understanding of the core ideas. No calculations required.
A.1 Explain in your own words why the electron cannot literally be "spinning." Give at least two independent arguments — one based on special relativity and one based on the pointlike nature of the electron.
A.2 The Stern-Gerlach experiment with silver atoms produces exactly two spots. Why can this result not be explained by orbital angular momentum alone, regardless of the value of $\ell$?
A.3 A student claims: "Spin-up along $z$ means the spin vector points in the $+z$ direction." Explain what is wrong with this statement. What does spin-up along $z$ mean, precisely?
A.4 In the three-stage Stern-Gerlach experiment ($z$-up $\to$ $x$-up $\to$ $z$-measurement), the final measurement gives 50% spin-up and 50% spin-down. A classically minded friend says: "The atoms must have carried both $S_z$ and $S_x$ information all along — we just didn't look carefully enough." How would you refute this argument using the experimental results?
A.5 Why does a spin-1/2 particle require a $4\pi$ ($720°$) rotation to return to its original state, rather than the $2\pi$ ($360°$) required for ordinary vectors? Is this a mathematical curiosity or a physically measurable effect?
A.6 The Bloch sphere maps the two-dimensional complex Hilbert space of spin-1/2 to the surface of a sphere in three-dimensional real space. How many real parameters describe a normalized spin-1/2 state (ignoring overall phase), and why does this match the dimensionality of a sphere surface?
A.7 Explain why $\langle\hat{S}_z\rangle$ is constant during Larmor precession in a $z$-directed magnetic field, even though $\langle\hat{S}_x\rangle$ and $\langle\hat{S}_y\rangle$ oscillate. Connect your answer to the commutator $[\hat{H}, \hat{S}_z]$.
A.8 A photon has spin-1, yet its spin component along its direction of travel can only be $\pm\hbar$ (not $0$). What is special about massless particles that eliminates the $m_s = 0$ state?
Part B: Applied Problems (two stars)
These problems require direct application of the chapter's key equations.
B.1: Pauli Matrix Properties
(a) Verify explicitly that $\sigma_x \sigma_y = i\sigma_z$ by performing the matrix multiplication.
(b) Compute $(\hat{n}\cdot\boldsymbol{\sigma})^2$ for an arbitrary unit vector $\hat{n} = (\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)$ and show that the result is the identity matrix $I$.
(c) Using the result of (b), prove that $e^{i\alpha\,\hat{n}\cdot\boldsymbol{\sigma}} = \cos\alpha\,I + i\sin\alpha\,(\hat{n}\cdot\boldsymbol{\sigma})$.
B.2: Eigenstates of $\hat{S}_x$ and $\hat{S}_y$
(a) Find the eigenstates of $\hat{S}_x$ by solving the eigenvalue equation $\sigma_x|\chi\rangle = \pm|\chi\rangle$. Normalize your results.
(b) Express the eigenstates of $\hat{S}_z$ in terms of the eigenstates of $\hat{S}_x$.
(c) A spin-1/2 particle is in the state $|+\rangle_y$. What is the probability of measuring $S_x = +\hbar/2$? What about $S_x = -\hbar/2$?
B.3: General Spin Direction
A spin-1/2 particle is in the spin-up eigenstate along the direction $\hat{n} = (\sin 60°\cos 30°,\;\sin 60°\sin 30°,\;\cos 60°)$.
(a) Write the spinor $|\chi\rangle$ in the $\{|+\rangle, |-\rangle\}$ basis. (Use $\theta = 60°$, $\phi = 30°$.)
(b) What is the probability of measuring $S_z = +\hbar/2$?
(c) What is the probability of measuring $S_z = -\hbar/2$?
(d) Compute $\langle\hat{S}_x\rangle$, $\langle\hat{S}_y\rangle$, and $\langle\hat{S}_z\rangle$.
(e) Verify that $\langle\hat{S}_x\rangle^2 + \langle\hat{S}_y\rangle^2 + \langle\hat{S}_z\rangle^2 = (\hbar/2)^2$.
B.4: Stern-Gerlach Deflection
A beam of potassium atoms (mass $m_K = 6.49 \times 10^{-26}$ kg, one unpaired electron in the 4s orbital) with velocity $v = 600$ m/s passes through a Stern-Gerlach apparatus with magnetic field gradient $B' = 800$ T/m over a length $L = 0.08$ m.
(a) Calculate the force on the spin-up and spin-down atoms. Use $g_s = 2$.
(b) Calculate the vertical deflection of each beam after passing through the magnet.
(c) What is the separation between the two spots on the detector, assuming the detector is immediately after the magnet?
(d) If the detector is placed a distance $D = 0.5$ m downstream of the magnet, what is the total separation?
B.5: Sequential Stern-Gerlach with Arbitrary Angle
A spin-1/2 particle is prepared in the state $|+\rangle$ (spin-up along $z$) and then passes through an SG apparatus oriented at angle $\theta$ from the $z$-axis in the $xz$-plane.
(a) What is the probability of measuring $S_n = +\hbar/2$ as a function of $\theta$?
(b) Plot $P(+)$ and $P(-)$ as functions of $\theta$ from $0$ to $\pi$.
(c) If the particle passes through the $\theta$-filter (selecting $S_n = +\hbar/2$), it then enters a final SG apparatus along $z$. What is the probability of measuring $S_z = +\hbar/2$ in the final apparatus?
(d) For what value of $\theta$ does the three-stage experiment give a 50-50 outcome in the final measurement?
B.6: Larmor Precession
An electron is placed in a uniform magnetic field $\mathbf{B} = 0.5\,\hat{z}$ T. At $t = 0$, the electron is in the state $|+\rangle_x$ (spin-up along $x$).
(a) What is the Larmor frequency $\omega_0 = g_s eB_0/(2m_e)$? (Use $g_s = 2$.)
(b) Write the state $|\chi(t)\rangle$ as a function of time.
(c) Calculate $\langle\hat{S}_x\rangle(t)$, $\langle\hat{S}_y\rangle(t)$, and $\langle\hat{S}_z\rangle(t)$.
(d) At what time does the spin first point in the $+y$ direction? What is the probability of measuring $S_y = +\hbar/2$ at that time?
(e) At what time does the spin first point in the $-x$ direction?
B.7: Spin Precession About an Arbitrary Axis
A spin-1/2 particle is initially in the state $|+\rangle$ and is placed in a magnetic field $\mathbf{B} = B_0(\sin\alpha\,\hat{x} + \cos\alpha\,\hat{z})$, which lies in the $xz$-plane at angle $\alpha$ from the $z$-axis.
(a) Write the Hamiltonian $\hat{H} = -\gamma\hat{\mathbf{S}}\cdot\mathbf{B}$ as a $2\times 2$ matrix.
(b) Find the energy eigenvalues and eigenstates.
(c) For $\alpha = \pi/4$, determine the probability of measuring $S_z = +\hbar/2$ as a function of time.
B.8: Bloch Sphere Distances
(a) Show that the fidelity between two spin-1/2 states $|\chi_1\rangle$ and $|\chi_2\rangle$ on the Bloch sphere is $F = |\langle\chi_1|\chi_2\rangle|^2 = (1 + \hat{n}_1\cdot\hat{n}_2)/2$, where $\hat{n}_1$ and $\hat{n}_2$ are the Bloch vectors.
(b) Two states are separated by angle $\Theta$ on the Bloch sphere. What is the minimum $\Theta$ for which the states can be reliably distinguished (define "reliably" as fidelity $\leq 0.1$)?
(c) An experimentalist can resolve spin states with fidelity $F = 0.95$. What is the corresponding angle between the Bloch vectors?
Part C: Advanced/Synthesis Problems (three stars)
These problems require deeper analysis and synthesis of multiple concepts.
C.1: Uncertainty Relations for Spin
(a) For a spin-1/2 particle in the state $|+\rangle$, calculate $\Delta S_x$, $\Delta S_y$, and $\Delta S_z$.
(b) Verify the uncertainty relation $\Delta S_x \cdot \Delta S_y \geq \frac{1}{2}|\langle[\hat{S}_x, \hat{S}_y]\rangle|$ for this state.
(c) For a state $|\chi\rangle = \cos(\theta/2)|+\rangle + \sin(\theta/2)|-\rangle$, compute $\Delta S_x \cdot \Delta S_y$ as a function of $\theta$ and determine the value of $\theta$ that minimizes this product. Is the minimum consistent with the uncertainty relation?
(d) Show that for spin-1/2, the minimum uncertainty state for $\hat{S}_x$ and $\hat{S}_y$ is any eigenstate of $\hat{S}_z$, but the minimum uncertainty state for $\hat{S}_x$ and $\hat{S}_z$ is any eigenstate of either $\hat{S}_x$ or $\hat{S}_z$.
C.2: Spin Operator Along $\hat{n}$: Complete Analysis
For the operator $\hat{S}_n = \hat{n}\cdot\hat{\mathbf{S}}$ where $\hat{n} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)$:
(a) Write $\hat{S}_n$ as a $2\times 2$ matrix.
(b) Find the eigenvalues of $\hat{S}_n$. (You should get $\pm\hbar/2$ regardless of $\hat{n}$.)
(c) Find the normalized eigenstates $|+\rangle_n$ and $|-\rangle_n$.
(d) Verify that $|+\rangle_n$ and $|-\rangle_n$ are orthogonal.
(e) Show that $|+\rangle_n\langle+|_n + |-\rangle_n\langle-|_n = I$ (completeness).
(f) Express $\hat{S}_n$ in terms of its eigenstates as $\hat{S}_n = \frac{\hbar}{2}(|+\rangle_n\langle+|_n - |-\rangle_n\langle-|_n)$.
C.3: Spin-1 Stern-Gerlach
A spin-1 particle passes through a Stern-Gerlach apparatus oriented along $z$, producing three beams corresponding to $m_s = +1, 0, -1$.
(a) The $m_s = +1$ beam is selected and passes through a second SG apparatus oriented at angle $\theta$ from $z$ in the $xz$-plane. Using the spin-1 rotation matrix, find the probabilities of measuring $m_s' = +1, 0, -1$ along the new axis.
(b) Show that for $\theta = \pi/2$ (measurement along $x$), the probabilities are $P(+1) = 1/4$, $P(0) = 1/2$, $P(-1) = 1/4$.
(c) For spin-1, construct the operator $\hat{S}_x^2$ and find its eigenvalues. Show that unlike spin-1/2, spin-1 has a state with zero angular momentum along a given axis — the $m_s = 0$ state.
C.4: The Density Matrix for Spin-1/2
(a) For a pure state $|\chi\rangle = \alpha|+\rangle + \beta|-\rangle$, write the density matrix $\hat{\rho} = |\chi\rangle\langle\chi|$ as a $2\times 2$ matrix.
(b) Express the density matrix in terms of the Pauli matrices: $\hat{\rho} = \frac{1}{2}(I + \mathbf{r}\cdot\boldsymbol{\sigma})$, where $\mathbf{r}$ is the Bloch vector. Show that $|\mathbf{r}| = 1$ for pure states.
(c) For a maximally mixed state $\hat{\rho} = I/2$, what is the Bloch vector? Interpret this physically.
(d) An ensemble of spin-1/2 particles has 70% in state $|+\rangle$ and 30% in state $|-\rangle$. Compute the density matrix, the Bloch vector, and $\langle\hat{S}_z\rangle$.
(e) Show that $\text{Tr}(\hat{\rho}^2) = (1 + |\mathbf{r}|^2)/2$ and hence $\text{Tr}(\hat{\rho}^2) = 1$ for pure states and $\text{Tr}(\hat{\rho}^2) < 1$ for mixed states.
C.5: Neutrino Spin Helicity
Neutrinos are spin-1/2 particles that (in the Standard Model) are always observed to be left-handed: their spin is antiparallel to their momentum. Antineutrinos are always right-handed.
(a) Define the helicity operator $\hat{h} = \hat{\mathbf{S}}\cdot\hat{\mathbf{p}}/|\hat{\mathbf{p}}|$ and show that its eigenvalues are $\pm\hbar/2$.
(b) For a neutrino traveling in the $+z$ direction with helicity $h = -\hbar/2$, write the spin state in the $\{|+\rangle, |-\rangle\}$ basis.
(c) If the same neutrino is now analyzed by an SG apparatus oriented along the $x$-axis, what are the probabilities of finding $S_x = \pm\hbar/2$?
(d) Can a massive spin-1/2 particle have definite helicity in all reference frames? What does this imply about neutrino mass?
C.6: NMR Spin Echo
In nuclear magnetic resonance, a proton spin precesses in a strong field $B_0\hat{z}$. At $t = 0$, a resonant RF pulse rotates the spin by $\pi/2$ about the $x$-axis (a "$\pi/2$ pulse"), placing it in the equatorial plane of the Bloch sphere.
(a) If the spin starts as $|+\rangle$, what is the state immediately after the $\pi/2$ pulse? (A rotation by $\pi/2$ about $\hat{x}$: $\hat{R}_x(\pi/2) = e^{-i\pi\sigma_x/4}$.)
(b) The spin then precesses freely for time $\tau$. Write the state at $t = \tau$.
(c) At $t = \tau$, a $\pi$ pulse about $\hat{x}$ is applied. Show that this pulse effectively reverses the accumulated phase.
(d) After another free precession period $\tau$, show that the spin returns to its state immediately after the first pulse (the "spin echo"). This is the Hahn echo, the foundation of MRI contrast.
C.7: Magnetic Resonance: Rotating Frame
A spin-1/2 particle is in a static field $B_0\hat{z}$ plus a rotating transverse field $\mathbf{B}_1(t) = B_1(\cos\omega t\,\hat{x} + \sin\omega t\,\hat{y})$.
(a) Write the full Hamiltonian $\hat{H}(t)$.
(b) Transform to the frame rotating at frequency $\omega$ about $\hat{z}$. Show that in this frame, the effective Hamiltonian is time-independent:
$$\hat{H}_{\text{rot}} = -\frac{\hbar}{2}\left[(\omega_0 - \omega)\sigma_z + \omega_1 \sigma_x\right]$$
where $\omega_0 = \gamma B_0$ and $\omega_1 = \gamma B_1$.
(c) At resonance ($\omega = \omega_0$), the effective Hamiltonian is purely along $\hat{x}$. Show that a spin initially along $+z$ will oscillate between $|+\rangle$ and $|-\rangle$ at the Rabi frequency $\omega_1$.
(d) What is the condition for a "$\pi$ pulse" — a pulse that flips the spin from $|+\rangle$ to $|-\rangle$?
C.8: Projective Measurement Statistics
An experimentalist performs $N = 1000$ measurements of $S_z$ on identically prepared spin-1/2 particles, each in the state $|\chi\rangle = \cos(20°)|+\rangle + \sin(20°)|-\rangle$ (with $\theta = 40°$ on the Bloch sphere).
(a) What are the theoretical probabilities $P(+)$ and $P(-)$?
(b) What is the expected number of spin-up and spin-down results?
(c) What is the standard deviation of the number of spin-up results? (Binomial statistics: $\sigma = \sqrt{NP(+)P(-)}$.)
(d) If the experimentalist instead measures $S_x$, what are the probabilities? (You'll need to express $|\chi\rangle$ in the $\hat{S}_x$ eigenbasis.)
C.9: Spin and the Hydrogen Atom
The ground state of hydrogen has $n = 1$, $\ell = 0$, $m_\ell = 0$. Including electron spin, the full state is $|n, \ell, m_\ell\rangle \otimes |m_s\rangle$.
(a) How many states exist in the ground level when spin is included? List them all.
(b) In the first excited level ($n = 2$), how many states are there when spin is included? List the quantum numbers $(\ell, m_\ell, m_s)$ for all states.
(c) In a magnetic field $\mathbf{B} = B_0\hat{z}$, the interaction Hamiltonian is $\hat{H}_B = -(\hat{\boldsymbol{\mu}}_L + \hat{\boldsymbol{\mu}}_S)\cdot\mathbf{B}$. For the ground state ($\ell = 0$), show that only the spin magnetic moment contributes, and find the energy splitting between the two ground-state levels.
(d) This splitting is the basis of the 21 cm hydrogen line (when the proton spin is also included). Explain qualitatively why the hyperfine splitting is much smaller than the fine structure splitting.
C.10: Constructing the Spin-3/2 Matrices
Using the general angular momentum matrix elements from Chapter 12:
(a) Construct the $4 \times 4$ matrices for $\hat{S}_z$, $\hat{S}_+$, $\hat{S}_-$, $\hat{S}_x$, and $\hat{S}_y$ for spin-3/2.
(b) Verify that $\hat{S}^2 = \frac{15}{4}\hbar^2 I$.
(c) Verify the commutation relation $[\hat{S}_x, \hat{S}_y] = i\hbar\hat{S}_z$.
(d) Find the eigenvalues of $\hat{S}_x$ and verify they are $\pm\frac{3}{2}\hbar$ and $\pm\frac{1}{2}\hbar$.
Part D: Computational Problems (two to three stars)
These problems are best solved with Python. Use the toolkit module from code/project-checkpoint.py.
D.1: Stern-Gerlach Cascade (two stars)
Simulate a cascade of Stern-Gerlach devices: - SG along $z$ (select up) - SG along $\hat{n}_1 = (\sin\theta_1, 0, \cos\theta_1)$ (select up) - SG along $z$ (measure)
Plot the probability of final spin-up as a function of $\theta_1$ from $0$ to $\pi$. Verify the analytical result $P(+) = \cos^4(\theta_1/2) + \sin^4(\theta_1/2)$.
D.2: Precession Visualization (two stars)
Create an animation (or a sequence of Bloch sphere plots at different times) showing Larmor precession for a spin initially at $\theta = \pi/3$, $\phi = 0$ in a field $\mathbf{B} = B_0\hat{z}$. Show one complete precession period. Include the trajectory as a colored line that fades with time.
D.3: Spin-1 Measurement Probabilities (three stars)
For a spin-1 particle initially in the $|1, +1\rangle$ state, compute and plot the probabilities $P(m_s' = +1, 0, -1)$ as functions of the measurement angle $\theta$ (angle between measurement axis and $z$). Verify the known result:
$$P(+1) = \cos^4(\theta/2), \quad P(0) = \frac{1}{2}\sin^2\theta, \quad P(-1) = \sin^4(\theta/2)$$
D.4: Random Spin State Ensemble (three stars)
Generate $N = 10000$ uniformly random pure spin-1/2 states on the Bloch sphere. For each state, compute $\langle\hat{S}_z\rangle$. Plot a histogram of $\langle\hat{S}_z\rangle$ values and verify that the distribution is uniform on $[-\hbar/2, +\hbar/2]$. Explain why this is the case geometrically.
Solutions Guide
Selected answers for self-checking (full solutions in the instructor guide):
- A.2: Integer $\ell$ gives $2\ell + 1$ spots (always odd). Two spots requires half-integer $j = 1/2$.
- B.2(c): $P(S_x = +\hbar/2) = 1/2$ and $P(S_x = -\hbar/2) = 1/2$.
- B.3(b): $P(+) = \cos^2(30°) = 3/4$.
- B.5(a): $P(+) = \cos^2(\theta/2)$.
- B.6(a): $\omega_0 = eB_0/m_e \approx 8.79 \times 10^{10}$ rad/s.
- B.6(d): $t = \pi/(2\omega_0)$; $P(S_y = +\hbar/2) = 1$.
- C.1(a): $\Delta S_x = \Delta S_y = \hbar/2$, $\Delta S_z = 0$.
- C.3(b): $P(+1) = 1/4$, $P(0) = 1/2$, $P(-1) = 1/4$ for $\theta = \pi/2$.
- C.9(a): Two states: $|1, 0, 0\rangle \otimes |+\rangle$ and $|1, 0, 0\rangle \otimes |-\rangle$.
- C.9(b): $2 \times (1 + 3) = 8$ states total.