Born Rule & Wavefunction Collapse

Prepare the Ensemble

Mixing angle θ 90°

Relative phase φ 0°

Ensemble size N 400

Quantum State (before measurement)

|ψ⟩ = 0.71|0⟩ + 0.71|1⟩

Born-rule prediction

P(|0⟩) = 50.0%P(|1⟩) = 50.0%

Actual outcome frequency

f(|0⟩) = —f(|1⟩) = —

Trials so far

|Δ| from Born rule

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How it works

Every one of the N spheres below starts prepared in the identical qubit superposition |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩, orbiting a shared point — a picture of coherent superposition, not N different particles.

Born rule:  P(0) = cos²(θ/2)
            P(1) = sin²(θ/2)
Collapse:   each trial samples one outcome from P(0)/P(1);
            the state instantly becomes |0⟩ or |1⟩ — no in-between.

Pressing Measure All N performs N *independent* measurements: every sphere independently samples an outcome with Math.random() weighted by its own P(0), then snaps — collapses — to the |0⟩ well (blue, left) or the |1⟩ well (orange, right). No single measurement ever tells you θ; only the outcome frequency over many trials approaches the true probability, exactly as the law of large numbers predicts.

θ slider — sets the amplitudes and hence the true Born-rule probabilities (θ=90° is a fair 50/50 superposition, θ=0°/180° are the definite states |0⟩/|1⟩).
φ slider — the relative phase between the two amplitudes; it changes the superposition's orbital orientation but — crucially — never changes the measurement probabilities, since Born-rule probabilities only depend on |amplitude|².
Re-prepare — returns every sphere to the uncollapsed superposition cloud, ready for a fresh measurement.
Auto-repeat — continuously re-prepares and re-measures the whole ensemble, accumulating trials so you can watch the running frequency converge on the theoretical curve.

Real-world relevance: this is exactly how a quantum-computer benchmark works — you cannot read out an amplitude directly, so you run the same circuit thousands of times and histogram the bitstrings, exactly as this ensemble does.

Collapsed to |0⟩ Collapsed to |1⟩ Superposition (uncollapsed)

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Born Rule & Wavefunction Collapse: Ensemble Measurement Simulator