The state is a complex amplitude ψ. You never see ψ — you see where the particle lands, and the odds of landing at x are |ψ(x)|². Born's rule. That squared magnitude is a genuine probability density: non-negative everywhere, normalized so the total is 1, and—because amplitudes add before they are squared—it carries an interference cross-term no bag of classical odds can. Down the center, a packet goes in, the engine integrates |ψ|², and the proven density comes out. The blue team builds it; the red team breaks it.
source Born, Zur Quantenmechanik der Stoßvorgänge (1926), Z. Phys. 37, 863–867 — the probability rule — doi:10.1007/BF01397477 AMBER (paywalled; no free stable full text). Rendered, not quoted.
Three things make |ψ|² a probability density, and all three are checked live on the current state:
P1 non-negative everywhere (a magnitude squared cannot be <0). P2 normalized — ∫|ψ|² dx = 1. P3 expectations follow: ⟨x⟩ = ∫ x|ψ|² dx. Superposition adds amplitudes, then squares — the cross-term is the interference.
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Schrödinger gave the equation whose solution is ψ, a complex amplitude — and no one knew what it meant. Born, 1926, supplied the meaning in one footnote: the physical thing is |ψ|², the probability density.
So the state the-schrödinger-equation evolves is not itself observable; its squared magnitude is. Each sphere is the next one's premise — the wave here becomes the odds.
The blue team's live check: re-integrate the density ∫|ψ|² dx over the current state and confirm it equals 1. If red tampers (uses |ψ| instead of |ψ|²), the total is no longer 1 — this badge is where it shows.
Feed the engine a state. A Gaussian wavepacket is ψ(x) = (2πσ²)−1/4 e−(x−x₀)²/4σ² ei k x — width σ, momentum k. In superposition mode you feed two packets, ψ₁+ψ₂, separated by d.
Natural units: ħ = m = 1. Only the amplitude goes in; the engine turns it into odds.
Single Gaussian: one hump, ⟨x⟩=0, no interference. Switch to superposition to see the cross-term.
Every number is a live numeric integral of |ψ|² over the grid — never looked up.
What the machine produces, proven: a normalized probability density. ∫|ψ|² dx = 1.000000, ⟨x⟩ = 0, and in superposition an interference cross-term of — — so |ψ₁+ψ₂|² ≠ |ψ₁|²+|ψ₂|². Orthonormal box modes give ∫φmφn = δmn.
The blue team's witness (left) confirms the total is 1 live; the red team (right) tries to make it otherwise.
And |ψ|² gives probabilities, not outcomes: the measurement problem — when and how one result is selected — is unsolved. For N particles ψ lives in 3N-dimensional configuration space, not real 3-space. The rule is exact and it works; its meaning is a choice (Copenhagen / many-worlds / pilot-wave), which is why this is one frame, not the whole story.
"The probability is |ψ|." Cut. Born's main text first wrote it proportional to ψ; the square appeared in a footnote added in proof. It is |ψ|² — the tamper (window 6) revives the wrong version and the witness kills it.
"ψ is a real wave in space." Cut. For many particles it is a wave in high-dimensional configuration space; only |ψ|² is observable, and only as statistics over repeats.
"Probability just means our ignorance." Kept, corrected. The nonzero cross-term — computed here — is amplitude interference, not classical odds; Bell/PBR bound any "hidden ignorance" story.
The red team's move: use |ψ| (not |ψ|²) as the density — Born's discarded first draft. The state is still normalized by its true |ψ|² norm, so the total no longer sums to 1.
Swap |ψ|² for |ψ| and ∫(density) drifts off 1 — the witness (window 7) recomputes, disagrees with 1, and turns red. Nothing is faked; the attack is real and it is caught.