Born's rule, density and normalisation

For an observable with a non-degenerate discrete spectrum, Born's rule is |⟨aₙ|ψ⟩|²: the squared modulus of the component along the eigenvector. Position breaks that recipe — x̂'s spectrum is the whole real line and it has no eigenvectors in L²(ℝ), since ⟨x|x′⟩ = δ(x − x′) is no norm at all. The spectral theorem repairs it: to every interval [a, b] it assigns a projector Ê([a, b]), which in the position representation simply multiplies ψ by the interval's indicator function, so P(a < x < b) = ⟨ψ|Ê([a, b])|ψ⟩ = ∫ₐᵇ|ψ(x)|²dx. Read backwards, that makes |ψ(x)|² a probability density — probability per unit length, meaningful only under an integral — and it is why a continuous spectrum attaches probability to intervals while a discrete one attaches it to eigenvalues.

A probability is dimensionless, so ∫|ψ|²dx = 1 forces [|ψ|²] = m⁻¹ and [ψ] = m⁻¹ᐟ² in one dimension; in three, ∫|ψ|²d³r = 1 makes it m⁻³ᐟ². The infinite well's ground state peaks at |ψ₁(L/2)|² = 2/L — for L = 1.0 nm that is 2.0 nm⁻¹, or 2.0 × 10⁹ m⁻¹ in SI, the number swinging by nine orders of magnitude under a unit change no probability could survive. Discrete expansion coefficients behave differently: cₙ = ⟨n|ψ⟩ against an orthonormal eigenbasis is dimensionless, and |cₙ|² is already a probability with no integral needed. Track which of the two you are holding, and dimensional analysis will catch most Born-rule errors before the arithmetic starts.

Solving an eigenvalue problem returns ψ only up to scale, so fix it. A Gaussian ψ(x) = N e^{(−x²/4σ²)} has ∫|N|²e^{(−x²/2σ²)}dx = |N|²σ√(2π), and ⟨ψ|ψ⟩ = 1 gives N = (2πσ²)^(−1/4) — about 0.63 nm⁻¹ᐟ² for σ = 1.0 nm. Only the modulus is fixed: N e^(iα) works equally well for any real α, which is the precise sense in which a state is a ray rather than a vector. And the job is done once. The Schrödinger equation gives d⟨ψ|ψ⟩/dt = (i/ħ)⟨ψ|(H† − H)|ψ⟩, which vanishes exactly because H is Hermitian — self-adjointness is not a technicality but the guarantee that today's normalisation still holds at every later time.

Take the n = 2 well state ψ₂ = √(2/L) sin(2πx/L) and ask for the probability at exactly x = L/4, its antinode: the honest answer is zero, because an integral over a width-zero interval vanishes, node or antinode alike. What a point value does buy is a ratio and an estimate. The density at L/4 is 2/L while at L/8 it is 1/L, so equal narrow windows around those points collect probability 2:1, and a window of width 0.010L at L/4 holds ≈ (2/L)(0.010L) = 0.020. The estimate is only as good as the window is narrow against the scale on which the density changes: stretch it to a sizeable fraction of the L/2 spacing between nodes and only the integral will do.

e^(ikx) solves the free eigenvalue problem, yet |e^(ikx)|² = 1 makes ⟨ψ|ψ⟩ = ∫1 dx diverge: momentum eigenfunctions are not vectors of L²(ℝ), and Born's rule says nothing about them directly. Two conventions rescue the calculation. Box normalisation confines the wave to a length L_box with periodic boundary conditions, giving ψₖ = e^(ikx)/√L_box, discrete k = 2πn/L_box and a uniform density 1/L_box, with L_box cancelling from any physical answer at the end. Delta normalisation keeps the line infinite and sets ⟨k|k′⟩ = δ(k − k′) via ψₖ = e^(ikx)/√(2π), trading probabilities for densities per unit k. Both are scaffolding: every preparable state is a normalisable wave packet, and a "plane wave" is shorthand for a packet much longer than anything else in the problem.

Discretise ψ on a grid of spacing dx and every statement above becomes a few lines of NumPy. The norm is the Riemann sum dx Σⱼ|ψⱼ|², standing in for ⟨ψ|ψ⟩; dividing ψ by the square root of that sum normalises it, and an interval probability is the same sum over a slice of the grid. The Born rule itself can be simulated: numpy.random.choice, given the grid positions and the weights |ψⱼ|²dx, draws positions one at a time, and a histogram of 10⁴ draws hugs |ψ|² with the relative scatter 1/√(counts per bin) of any counting experiment. Two checks keep the grid honest: the summed norm of an analytically normalised ψ should approach 1 as dx shrinks (trapezoid error ~ dx² for smooth ψ), and the box edges must sit where ψ has already decayed, or the grid silently renormalises probability that belongs outside.