QUANTUM LABORATORY / 01
Qubit Box
Prepare a quantum state. See what changes. Measure what remains.
|0⟩ ≡ n = 1 / |1⟩ ≡ n = 2
POSITION REPRESENTATION
Where could the particle be found?
Energy populations stay fixed. Relative phase changes the interference pattern.
STATE GEOMETRY
The Bloch sphere
A map of the two-level state, not a sphere in physical space. Mixed states lie inside.
MEASUREMENT
Prediction becomes a record
A single measurement changes this state. An ensemble samples independently prepared copies.
Same energy probabilities. Different quantum states.
Choose the incoherent mixture above. Both energy outcomes remain 50/50, but the interference disappears. Measure in X to distinguish these preparations.
Equations, interpretation & model scope +
The physical model
An infinite one-dimensional square well, restricted to its first two energy eigenstates. Here ℏ = m = L = 1; Eₙ = n²π²/2 and ψₙ(x) = √2 sin(nπx).
ρ = ½(I + r · σ)
P(x) = ρ₀₀ψ₁² + ρ₁₁ψ₂² + 2 Re(ρ₀₁)ψ₁ψ₂
P(± | a) = ½(1 ± r · a)
The coherence control scales the off-diagonal entries while preserving populations and positivity. It prepares a mixed state; it is not an environmental time-evolution solver.
What a measurement means here
Z measures energy. X and Y are ideal projective measurements within the encoded two-level space; implementing them physically requires basis-changing control. They are not position measurements. No position collapse is simulated because it generally populates higher box levels.
Evolution is analytic, with relative phase φ(t) = φ(0) − (E₂−E₁)t/ℏ. One beat period T = 2πℏ/(E₂−E₁). Playback is adjustable from 0.12 to 0.96 periods per second; the default 4× setting runs at 0.48 periods per second. Measurements pause evolution so repeated measurements in the same basis can be compared without intervening dynamics.
Ensemble results retain the frozen state and basis used for that batch. Their 95% Wilson interval is a fixed-sample summary, not an anytime hypothesis test.
References: Feynman · time evolution · IBM Quantum · measurements.