Born Rule & Wavefunction Collapse: 2D Bloch-Disk Ensemble Measurement

A single quantum measurement never reveals a probability — it only ever returns one of a handful of definite outcomes. This 2D simulator makes that statistical nature visible by preparing an entire ensemble of hundreds of identical qubits in the same superposition |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩ — set either with sliders or by dragging a point on a flattened Bloch-disk projection — then measuring all of them at once. Each dot independently collapses to |0⟩ or |1⟩ according to the Born-rule probabilities |α|² and |β|², and the running histogram of outcomes converges on the theoretical prediction as the sample size grows: a direct, hands-on demonstration of how repeated measurement, not a single reading, is what turns quantum amplitudes into the classical statistics we can actually observe.