Spin and the Pauli exclusion principle

Equipment

• Virtual Bloch sphere with a red state vector precessing in a field
• Sliders for polar angle θ, azimuth φ, time t
• Stern–Gerlach beam-splitting simulation with two spots
• Slider for the angle between preparation and measurement axes; particle count

Procedure

1. Read probabilities on the Bloch sphere

   In figure 1 set θ = 0: the vector points to the north pole and the readout gives P(+z) = 100%. Raise θ to 90° (equator) for P(+) = P(−) = 50%, and θ = 180° for P(−z) = 100% — matching $P(+)=\cos^2(\theta/2)$.

2. Analyze the Stern–Gerlach split

   In figure 2, set the axis angle to 0 and raise the particle count: all particles land in one spot. Set 90°: two equally bright spots. Read the 'Born prediction' line and compare the theoretical P(+) with the counted split.

3. Change precession time and conclude

   Raise the time slider t: the vector precesses around the field and ⟨σ⟩ rotates continuously, yet outcomes remain only ±ℏ/2 — the two-level nature of spin-1/2. Predict P(+) at θ = 60° and check with $P=\cos^2(30°)=0{,}75$.

Simulation

Experiment history

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