In this dictionary, the Born rule is the prescription that the probability of obtaining a particular measurement outcome equals the squared modulus of the amplitude (projection) of the system's wavefunction onto the outcome's eigenstate, after appropriate normalization.

Principle

Complex quantum amplitudes combine linearly; probabilities are obtained from amplitudes by taking the modulus squared, which maps complex amplitudes to nonnegative real numbers that sum to one under normalization.

Demonstration

In the double‑slit experiment the intensity distribution on a screen follows |ψ1+ψ2|^2, producing interference fringes whose relative frequencies match the squared amplitude; similarly, spin measurement probabilities follow |⟨spin‑basis|ψ⟩|^2 for each outcome.

Misapplication

Treating amplitudes themselves as probabilities, using unnormalized states without renormalization, or replacing the modulus‑squared by another power leads to inconsistent predictions; asserting the Born rule without specifying the measurement context (projective vs generalized measurement) causes misuse.

Consequence

When applied correctly, the Born rule allows quantitative statistical predictions of measurement outcomes, enables computation of expectation values and variances, and connects the formal wavefunction to experimentally observed frequencies.

Reversal

A reversal posits deterministic hidden variables or alternative mappings from amplitudes to probabilities (e.g., non‑standard probabilistic rules), which remove or alter the direct amplitude→probability relation of the Born rule.

Boundary

The Born rule applies within standard quantum mechanics to projective measurements and extends to generalized measurements (POVMs) via appropriate operator measures; it assumes a normalized state and a well‑specified measurement context and does not by itself resolve interpretational questions about collapse or ontology of the wavefunction.

Semantic Tension

Tension exists between viewing the Born rule as an empirical rule linking formal amplitudes to frequencies and attempts to derive it from deeper principles (decision theory, symmetries) or to replace it in alternative formulations; also between treating probabilities epistemically versus ontically.

Synthesis

The Born rule operationally bridges the mathematical amplitudes of quantum states and experimentally accessible probabilities: amplitudes interfere linearly, and the modulus‑squared rule yields nonnegative normalized probabilities appropriate for statistical prediction across measurements, subject to measurement context and normalization.