Case Study 1: Sequential Stern-Gerlach — A Complete Quantum Measurement Lab

Overview

The sequential Stern-Gerlach experiment is the single cleanest demonstration of quantum measurement in action. With nothing more than magnets, an oven, and a vacuum chamber, it reveals superposition, incompatible observables, state preparation, and the irreversible disturbance of measurement. This case study works through the experiment in exhaustive detail, treating the SG apparatus as a quantum measurement laboratory.

By the end, you will be able to predict the outcome of any sequence of Stern-Gerlach devices oriented along arbitrary axes — and you will understand why those predictions follow inevitably from the spin-1/2 formalism.

Part 1: The Single Stern-Gerlach Apparatus

The Physics

A beam of neutral silver atoms emerges from an oven with random spin orientations. Each atom carries a single unpaired electron in the $5s$ orbital ($\ell = 0$), so the magnetic moment is entirely due to electron spin:

$$\boldsymbol{\mu} = -g_s \frac{\mu_B}{\hbar}\hat{\mathbf{S}}, \qquad g_s \approx 2$$

The inhomogeneous magnetic field $\mathbf{B} \approx B_0\hat{z} + B'z\hat{z}$ exerts a force:

$$F_z = \mu_z \frac{\partial B}{\partial z} = \mp g_s\frac{\mu_B}{2}B'$$

Spin-up atoms ($m_s = +1/2$) are deflected in one direction; spin-down atoms ($m_s = -1/2$) in the other. The beam splits into exactly two components.

Quantum Mechanical Description

The state space of the spin degree of freedom is two-dimensional: $\mathcal{H}_{\text{spin}} = \text{span}\{|+\rangle, |-\rangle\}$.

An atom emerging from the oven with unknown spin is described by the density matrix $\hat{\rho}_{\text{initial}} = \frac{1}{2}I$ (maximally mixed state). The SG-$z$ apparatus performs a projective measurement of $\hat{S}_z$.

After the SG-$z$ apparatus, the two beams contain atoms in definite spin states:

• Upper beam: $|+\rangle$ (every atom has $S_z = +\hbar/2$ with certainty).
• Lower beam: $|-\rangle$ (every atom has $S_z = -\hbar/2$ with certainty).

By blocking one beam, the SG apparatus serves as a state preparation device. This is the most important conceptual point: measurement and preparation are two sides of the same coin.

Part 2: The Canonical Experiments

Experiment A: Same Axis ($z \to z$)

Setup: SG-$z$ selects $|+\rangle$. A second SG-$z$ measures the result.

Calculation:

$$P(S_z = +\hbar/2) = |\langle+|+\rangle|^2 = 1$$ $$P(S_z = -\hbar/2) = |\langle-|+\rangle|^2 = 0$$

Result: 100% spin-up. The measurement is repeatable.

Physical interpretation: A state prepared as an eigenstate of $\hat{S}_z$ remains an eigenstate of $\hat{S}_z$ when measured again along the same axis. There is no disturbance because the measurement operator commutes with the preparation.

Experiment B: Perpendicular Axes ($z \to x$)

Setup: SG-$z$ selects $|+\rangle$. SG-$x$ measures the result.

Calculation: We need $|+\rangle$ in the $\hat{S}_x$ eigenbasis:

$$|+\rangle = \frac{1}{\sqrt{2}}|+\rangle_x + \frac{1}{\sqrt{2}}|-\rangle_x$$

Therefore:

$$P(S_x = +\hbar/2) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$$ $$P(S_x = -\hbar/2) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$$

Result: 50-50 split.

Physical interpretation: An eigenstate of $\hat{S}_z$ has maximum uncertainty in $\hat{S}_x$. The information about $S_x$ is completely absent from the state $|+\rangle$ — it is not "hidden" or "unknown," it genuinely does not exist until measured.

Experiment C: The Three-Stage Experiment ($z \to x \to z$)

This is the experiment that makes classical physicists uncomfortable.

Setup: SG-$z$ selects $|+\rangle$. SG-$x$ selects $|+\rangle_x$. A third SG-$z$ measures the result.

Step 1: After SG-$z$, state is $|+\rangle$.

Step 2: The SG-$x$ measurement on $|+\rangle$ gives $|+\rangle_x$ with probability $1/2$ (we select only these atoms). The post-measurement state is:

$$|+\rangle_x = \frac{1}{\sqrt{2}}|+\rangle + \frac{1}{\sqrt{2}}|-\rangle$$

Step 3: The third SG-$z$ measures this state:

$$P(S_z = +\hbar/2) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$$ $$P(S_z = -\hbar/2) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$$

Result: 50% spin-up, 50% spin-down.

The classical impossibility: We started with atoms that were 100% spin-up along $z$. By measuring along $x$ (and selecting the $+x$ result), we "reset" the $z$-information. The $x$-measurement has irrevocably disturbed the $z$-component. This is not a failure of experimental precision — it is a fundamental feature of quantum mechanics, encoded in the non-commutativity $[\hat{S}_z, \hat{S}_x] = i\hbar\hat{S}_y \neq 0$.

Combined probabilities: Starting from the oven with an unprepared beam: - Probability of passing SG-$z$ (up): $1/2$ - Probability of passing SG-$x$ (up), given up from $z$: $1/2$ - Probability of measuring $z$-up at the end: $1/2$

Total probability of measuring $z$-up at the end: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.

Part 3: Arbitrary Angles

SG-$z \to$ SG-$\hat{n}(\theta)$

The most general two-stage experiment: prepare $|+\rangle$ with SG-$z$, then measure along $\hat{n} = (\sin\theta, 0, \cos\theta)$ (in the $xz$-plane for simplicity).

The eigenstate of $\hat{S}_n$ with eigenvalue $+\hbar/2$ is:

$$|+\rangle_n = \cos\frac{\theta}{2}|+\rangle + \sin\frac{\theta}{2}|-\rangle$$

The probability of measuring spin-up along $\hat{n}$:

$$P_n(+) = |\langle+|_n\;|+\rangle|^2 = \cos^2\frac{\theta}{2}$$

$$P_n(-) = \sin^2\frac{\theta}{2}$$

Special cases: | $\theta$ | Physical situation | $P(+)$ | $P(-)$ | |----------|-------------------|---------|---------| | $0$ | Same axis | $1$ | $0$ | | $\pi/6$ | 30° tilt | $0.933$ | $0.067$ | | $\pi/4$ | 45° tilt | $0.854$ | $0.146$ | | $\pi/3$ | 60° tilt | $0.750$ | $0.250$ | | $\pi/2$ | Perpendicular | $0.500$ | $0.500$ | | $2\pi/3$ | 120° tilt | $0.250$ | $0.750$ | | $\pi$ | Opposite axis | $0$ | $1$ |

Three-Stage with Arbitrary Middle Angle: $z \to \hat{n}(\theta) \to z$

Setup: SG-$z$ prepares $|+\rangle$. SG-$\hat{n}$ at angle $\theta$ selects the $+\hat{n}$ beam. Final SG-$z$ measures.

After the middle stage, the state is $|+\rangle_n = \cos(\theta/2)|+\rangle + \sin(\theta/2)|-\rangle$.

Final probabilities:

$$P(S_z = +\hbar/2) = \cos^2\frac{\theta}{2}$$ $$P(S_z = -\hbar/2) = \sin^2\frac{\theta}{2}$$

The probability that the particle passes the middle filter AND is found spin-up at the end:

$$P_{\text{total}}(+) = \cos^2\frac{\theta}{2} \times \cos^2\frac{\theta}{2} = \cos^4\frac{\theta}{2}$$

For $\theta = \pi/2$: $P_{\text{total}}(+) = (1/\sqrt{2})^4 = 1/4$. Adding the probability of "down-up": $\sin^2(\theta/2)\cos^2(\theta/2) = 1/4$. But wait — if we selected the $+\hat{n}$ beam, we already have $P(+) = \cos^2(\theta/2)$ for the final measurement. The total probability accounting for the middle filter is:

$$P(\text{pass middle AND final up}) = \cos^2\frac{\theta}{2} \cdot \cos^2\frac{\theta}{2} = \cos^4\frac{\theta}{2}$$

This is maximized at $\theta = 0$ (trivial — the middle filter is the same axis) and equals zero at $\theta = \pi$ (the middle filter rejects everything).

Part 4: Multiple Intermediate Stages

Can We Beat the Three-Stage Limit?

In the three-stage experiment ($z \to x \to z$), we lose information. Can we recover it by inserting more intermediate measurements?

Setup with $N$ intermediate stages: SG-$z$ prepares $|+\rangle$. Then $N$ SG measurements at angles $\theta_k = k\pi/(2N)$ for $k = 1, 2, \ldots, N$ (each selecting spin-up), followed by a final SG-$z$.

Each step rotates the measurement axis by a small angle $\Delta\theta = \pi/(2N)$. The probability of passing each stage is $\cos^2(\Delta\theta/2)$. The total probability of passing all $N$ stages and emerging as spin-up in $z$ is:

$$P_{\text{total}} = \left[\cos^2\frac{\pi}{4N}\right]^N$$

For large $N$:

$$\cos\frac{\pi}{4N} \approx 1 - \frac{\pi^2}{32N^2} + \ldots$$

$$P_{\text{total}} \approx \left(1 - \frac{\pi^2}{32N^2}\right)^N \to 1 \quad \text{as } N \to \infty$$

Remarkable result: By making many small rotations of the measurement axis, we can gradually rotate the spin from $+z$ to $+x$ to $-z$ to any direction — with probability approaching 1! Each small measurement barely disturbs the state, and the cumulative effect is a smooth rotation.

This is a preview of the quantum Zeno effect (which we will encounter in Chapter 28 on the measurement problem): sufficiently frequent measurements can freeze or guide a quantum system's evolution.

Part 5: The SG Experiment as a Quantum Gate

The connection between the Stern-Gerlach experiment and quantum computing is direct and deep. Each SG apparatus performs a specific quantum operation:

State Preparation

An SG-$z$ apparatus that blocks the lower beam prepares $|+\rangle = |0\rangle$ (in quantum computing notation). This is the initialization step of any quantum algorithm.

Measurement

An SG apparatus along axis $\hat{n}$ performs a projective measurement of $\hat{S}_n$. This is the readout step.

Unitary Rotation

A magnetic field region (without spatial separation) rotates the spin state:

$$|\chi\rangle \to e^{-i\omega t\,\hat{n}\cdot\boldsymbol{\sigma}/2}|\chi\rangle$$

This is a single-qubit gate. By choosing $\hat{n}$ and $\omega t$, we can implement any rotation on the Bloch sphere — and therefore any single-qubit unitary operation.

The Complete Quantum Gate Set

From SG components alone:

• $R_z(\phi)$ gate: Precession in $B_0\hat{z}$ for time $t = \phi/\omega_0$.
• $R_x(\phi)$ gate: Precession in $B_0\hat{x}$ for time $t = \phi/\omega_0$.
• Hadamard-like gate: $R_y(\pi/2)$ — rotation by $90°$ about $y$-axis.

Any single-qubit operation can be decomposed into these rotations (Euler angle decomposition). The Stern-Gerlach apparatus, conceived in 1921, contains all the ingredients of single-qubit quantum computing.

Part 6: Worked Problems

Problem 1: SG-$z$ (up) $\to$ SG-$y$ (up) $\to$ SG-$z$

Step 1: Start with $|+\rangle$.

Step 2: Express $|+\rangle$ in the $\hat{S}_y$ eigenbasis:

$$|+\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle + i|-\rangle), \qquad |-\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle - i|-\rangle)$$

Inverting: $|+\rangle = \frac{1}{\sqrt{2}}(|+\rangle_y + |-\rangle_y)$.

Probability of passing the SG-$y$ (up) filter: $P = |\langle+|_y|+\rangle|^2 = |1/\sqrt{2}|^2 = 1/2$.

After filtering, state is $|+\rangle_y = \frac{1}{\sqrt{2}}(|+\rangle + i|-\rangle)$.

Step 3: Measure $\hat{S}_z$:

$$P(S_z = +\hbar/2) = \left|\frac{1}{\sqrt{2}}\right|^2 = \frac{1}{2}$$

Result: 50-50, identical to the $z \to x \to z$ experiment. This is expected: the $y$-axis is also perpendicular to $z$.

Problem 2: SG-$z$ (up) $\to$ SG at $45°$ (up) $\to$ SG-$z$

Step 1: Start with $|+\rangle$.

Step 2: $\hat{n} = (\sin 45°, 0, \cos 45°)$, so $\theta = 45°$.

$$|+\rangle_n = \cos(22.5°)|+\rangle + \sin(22.5°)|-\rangle$$

Probability of passing the $45°$ filter: $P = \cos^2(22.5°) \approx 0.854$.

Step 3: Measure $\hat{S}_z$:

$$P(S_z = +\hbar/2) = \cos^2(22.5°) \approx 0.854$$ $$P(S_z = -\hbar/2) = \sin^2(22.5°) \approx 0.146$$

The intermediate measurement at $45°$ partially preserves the $z$-information. The closer $\theta$ is to zero, the more information is preserved.

Problem 3: Four-Stage Cascade

SG-$z$ (up) $\to$ SG at $30°$ (up) $\to$ SG at $60°$ (up) $\to$ SG at $90°$ (up) $\to$ SG-$z$

Each stage rotates the measurement axis by $30°$.

After SG at $30°$: State is $|+\rangle_{30°} = \cos(15°)|+\rangle + \sin(15°)|-\rangle$. Probability of passing: $\cos^2(15°) \approx 0.933$.

After SG at $60°$: We need the probability of passing this filter given the state $|+\rangle_{30°}$. The angle between the $30°$ axis and the $60°$ axis is $30°$, so $P = \cos^2(15°) \approx 0.933$.

After SG at $90°$ ($x$-axis): Again $30°$ rotation, so $P \approx 0.933$.

The state is now $|+\rangle_x = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)$.

Final SG-$z$: $P(+) = 1/2$.

Cumulative probability of reaching the end as spin-up: $0.933 \times 0.933 \times 0.933 \times 0.5 \approx 0.406$.

Compare with a single-stage $z \to x \to z$: probability of spin-up at end is $0.5 \times 0.5 = 0.25$. The gradual rotation preserves more of the original spin information.

Discussion Questions

1. In Experiment C (three-stage), does the $x$-measurement "destroy" the $z$-information, or is it more accurate to say that the $z$-information was never a well-defined property of the state $|+\rangle_x$?

2. The quantum Zeno effect (many intermediate measurements approaching identity) seems to violate the uncertainty principle — we appear to maintain precise knowledge of spin along a continuously changing axis. Is this a genuine violation, or does the uncertainty principle have something else to say?

3. If Stern and Gerlach had known about spin in 1922, would they have designed the experiment differently? What practical changes to the apparatus would improve the experiment's ability to discriminate between spin-1/2 and orbital angular momentum effects?

4. The sequential SG experiment is often used to teach quantum mechanics because it requires no knowledge of wavefunctions, differential equations, or infinite-dimensional Hilbert spaces — just $2 \times 2$ matrices. Is this simplicity a feature or a bug? What important aspects of quantum mechanics are invisible in the spin-1/2 formalism?

5. How would the three-stage experiment differ if we did not block the $|-\rangle_x$ beam in the middle stage, but instead recombined the two beams before the final SG-$z$? (This is the famous "recombination" variant. The answer is surprising.)