The Born rule and probability

Born's rule turns ⟨ψ|ψ⟩ = 1 from a convention into a physical requirement: the outcome probabilities are the squared moduli of expansion coefficients, and they total one exactly when the state does. Normalising fixes more than a constant. For ψ(x) = A e^(−|x|/a), the integral ∫|ψ|² dx = |A|²a forces A = 1/√a, and with it the dimension of every one-dimensional wavefunction: [ψ] = m⁻¹ᐟ², so that |ψ|² dx is a pure number. Two things escape the procedure. The overall phase does — e^(iα)|ψ⟩ has every squared overlap unchanged, which is why a state is a ray, not a vector. And the momentum eigenfunction e^(ikx) does — ∫|A|² dx diverges on the line, so no choice of A normalises it. Plane waves sit outside L² and take the delta convention ⟨p|p′⟩ = δ(p − p′) instead, and for them the Born rule delivers a probability density over p rather than a probability for any single sharp momentum.

Expand the state on the eigenbasis of the observable: |ψ⟩ = Σ cₙ|aₙ⟩ with cₙ = ⟨aₙ|ψ⟩, each coefficient a projection. The rule reads off P(aₙ) = |cₙ|² = cₙ*cₙ — the conjugate is compulsory, because c² of a complex number is complex and no probability is. Take |ψ⟩ = ((1+i)|a₁⟩ + 2|a₂⟩)/√6: the weights are |1+i|² = 2 and |2|² = 4, so P(a₁) = 1/3 and P(a₂) = 2/3, and Parseval's identity guarantees Σ|cₙ|² = ⟨ψ|ψ⟩ = 1 in any orthonormal basis, not just this one. Degeneracy changes the bookkeeping, not the rule: when the eigenvalue a spans a d-dimensional eigenspace, build the projector P̂ₐ = Σⱼ|a, j⟩⟨a, j| over it and P(a) = ⟨ψ|P̂ₐ|ψ⟩ — square first and then add over the orthogonal directions, never add the amplitudes. A continuous spectrum swaps the sum for an integral, P(a ∈ Δ) = ∫Δ |⟨a|ψ⟩|² da, where |⟨a|ψ⟩|² is now a density carrying the inverse dimension of a.

Probability is not just conserved in total; it moves locally. Subtract ψ × (Schrödinger)* from ψ* × (Schrödinger): the V terms cancel when V is real, and what survives is ∂ₜ|ψ|² + ∂ₓj = 0 with j = (ħ/m) Im(ψ*∂ₓψ) — the exact analogue of charge conservation, with |ψ|² as the charge. The current diagnoses states at a glance. Any real ψ has j = 0, which is why the bound states of a well sit still. A plane wave Ae^(ikx) carries j = |A|²ħk/m, density times velocity. Integrated over the line, the continuity equation reduces d⟨ψ|ψ⟩/dt to a boundary term, and L² states vanish at infinity, so the norm never drifts: normalise once at t = 0 and the dynamics — unitary because H is self-adjoint — preserves it forever. The exception proves the rule: give V a negative imaginary part, as absorbing boundaries in numerical scattering deliberately do, and the norm decays at the local rate (2|Im V|/ħ)|ψ|² — probability then leaves by construction, not by error.

Why the squared modulus and not some other function of the overlap? Ask what any probability assignment on a Hilbert space must satisfy: every projector gets a number in [0,1], orthogonal outcomes add, and each complete basis totals 1. Gleason's 1957 theorem answers: in dimension three or more, the only such measures are μ(P̂) = Tr(ρ̂P̂) for some density operator ρ̂ — for a pure state, exactly |⟨a|ψ⟩|². A rival exponent dies on contact: under a |c|⁴ rule the state (1,1,1)/√3 in ℂ³ would give each basis outcome (1/3)² = 1/9, totalling 1/3 rather than 1, and no renormalisation can fix every basis at once. The theorem needs d ≥ 3 because there each direction belongs to continuum-many orthonormal bases, and additivity across all of them is crushing. A lone qubit escapes: its orthogonal projectors come only in antipodal pairs on the Bloch sphere, additivity constrains each pair separately, and explicit non-Born measures can be written down. So the exponent 2 is forced everywhere except the one space too small for the argument to bite.

The rule predicts the statistics of an ensemble of identically prepared systems and refuses to say anything about a single run. That makes its test statistical: prepare N copies, count outcomes, and compare against Np with the binomial spread √(Np(1−p)). With P(E₂) = 0.64 and N = 10⁴ runs, expect 6400 ± 48 — a count of 6390 confirms the rule, and asking which runs would return E₂ is not a question the theory answers. What the statistics can distinguish is a superposition from a mixture. The superposition c₁ψ₁ + c₂ψ₂ has density |c₁|²|ψ₁|² + |c₂|²|ψ₂|² + 2 Re(c₁*c₂ψ₁*ψ₂); the mixture keeps only the first two terms. That cross term is the experimental fingerprint of coherence — it is what two-slit fringes measure and what decoherence destroys. What the postulate never supplies is a definition of measurement itself, or a mechanism selecting one outcome: that is the measurement problem, and it lives outside the rule, not inside it.

The Born rule gives numerical work its acceptance test. Sample ψ on a grid with spacing Δx and the norm becomes Σ|ψᵢ|²Δx — one line of NumPy, np.sum(np.abs(ψ)**2)*dx — and printing it every few steps is a habit worth building. Propagate with Crank–Nicolson, whose one-step map (I + iĤΔt/2ħ)⁻¹(I − iĤΔt/2ħ) is exactly unitary for Hermitian Ĥ: the norm then holds to machine rounding over thousands of steps. Propagate with explicit Euler, ψ → (I − iĤΔt/ħ)ψ, and the norm grows every step — the map's eigenvalues 1 − iEₙΔt/ħ all have modulus √(1 + (EₙΔt/ħ)²) > 1 — so the density inflates exponentially and every probability quietly becomes a lie. The printout has one blind spot: |ψᵢ|² never sees the phase e^(ikxᵢ), so a grid too coarse for the momenta present, with kΔx not small against 1, can hold a perfect norm while the dynamics go wrong — resolution needs its own test, such as halving Δx and comparing. Conservation of Σ|ψᵢ|²Δx is the discrete shadow of the continuity equation, and a scheme that breaks it has broken the probability interpretation itself.