Stern–Gerlach Sequential Measurements

Worked Example

Consider an initial spin-1/2 system in state |+z>. An intermediate measurement is performed along an axis n lying in the xz plane at angle θ from +z, followed by a final measurement along +z. Let θ = 90° and the operation be "Measure and forget".

1. Initial state: |ψ₀> = |+z>
2. Intermediate axis n: θ = 90°
3. Intermediate + probability: |<+n|+z>|² = cos²(θ/2) = cos²(45°) = 0.5
4. Intermediate - probability: |<-n|+z>|² = sin²(θ/2) = sin²(45°) = 0.5
5. Measure and forget implies a nonselective measurement. The final density matrix is a mixed state.
6. Final + probability given intermediate +: |<+z|+n>|² = cos²(45°) = 0.5
7. Final + probability given intermediate -: |<+z|-n>|² = sin²(45°) = 0.5
8. Final conditional + probability: 0.5 * 0.5 + 0.5 * 0.5 = 0.5

In this scenario, we obtain a final conditional +z probability of 0.5. Since we retain all branches (nonselective), the retained fraction is 1, and the joint retained and final + is also 0.5.

Experiment

Open the Stern–Gerlach Sequential Measurements module to interact with the apparatus.

1. Set the Intermediate axis degrees to 90°.
2. Set the Intermediate operation to Keep plus only.
3. Observe the intermediate + probability becomes 0.5, and the retained fraction drops to 0.5, because we discard the |−n> branch.
4. The final + conditional probability is 0.5, leaving a joint retained and final + probability of 0.25.
5. Compare this with setting the operation to No measurement, where the state remains undisturbed |+z> and the final + conditional probability is 1.

Limitations

The diagram does not represent magnetic precession evolution; this module isolates the effect of measurement back-action. It assumes ideal, instantaneous measurements described by projection operators, ignoring continuous measurement dynamics and non-ideal detector efficiencies.

References

For a detailed theoretical treatment of Stern-Gerlach experiments and sequential measurements, see:

• MIT 8.05 Sample Midterm Notes on Spin and Stern-Gerlach (PDF)