3Experiment 3

Quantum Measurement Statistics: Spin-½ Systems

Verify the Born-rule angle dependence for spin-½ and demonstrate that an intervening X-measurement scrambles the Z-outcome — a result with no classical analogue.

A. Born-rule angle dependence

P(+|θ_P, θ_M) = cos²((θ_P − θ_M)/2) — exact quantum-mechanical result for spin-½.

Prep angle θ_P (°)0°

Meas angle θ_M (°)90°

P(+) = cos²(Δθ/2)0.5000

P(−) = 1 − P(+)0.5000

Total trials: 0

0 (0.0% — theory 50.0%)

−

0 (0.0% — theory 50.0%)

B. Sequential measurement

Fixed protocol every trial: prepare |+z⟩, measure Z, thenmeasure X in between, then measure Z again.

Click a button to run the sequence.

With intervening X measurement: the X eigenstate has equal probability for either Z outcome (cos²(45°) = 0.5), so the second Z result is completely random. With no X measurement: the state is undisturbed and Z gives the same outcome.

Observations

Record your measured data in the tables below. Refer to the lab manual for the full procedure.

Table 3.1 — Born-rule angle dependence ( $θ_{P} = 0^{\circ}$ )

$θ_{M}$	Theory $P (+)$	Freq. run 1 (50 trials)	Freq. run 2 (50 trials)

Rows: θM=60°, θM=90°

Table 3.2 — Sequential measurement (200 trials each)

Intervening X	Match % (Step 3 = Step 1)

Rows: Off, On