Measurement, eigenstates and state update

An observable is represented by a Hermitian operator Â, and two of the three properties this lesson needs follow straight from that. Its eigenvalues are real, so a reading can be a number on a dial. Eigenvectors belonging to different eigenvalues are orthogonal, so distinct outcomes are perfectly distinguishable — no reading is ever ambiguous between two spectrum values. The third property, completeness, does not follow from Hermiticity alone; it is the spectral theorem for a self-adjoint operator that hands you Σₙ|aₙ⟩⟨aₙ| = 1̂, and every expansion written below quietly leans on it. Granted that, the spectrum — the whole list of aₙ — fixes in advance every number the apparatus is allowed to display, and nothing else can ever appear on the dial, in any state whatever. For a particle in a box of width L the energy menu is Eₙ = n²π²ħ²/2mL², isolated numbers with nothing in between, so a reading of 3.5E₁ is not merely unlikely but impossible. For x̂ the menu is the whole real line, which needs the continuous version of everything that follows. One detail matters later: a degenerate eigenvalue owns a subspace, not a single vector.

Given |ψ⟩ and the eigenbasis, one inner product per outcome does all the work: cₙ = ⟨aₙ|ψ⟩, which in one dimension is the overlap integral ∫ uₙ*(x) ψ(x) dx over the eigenfunctions uₙ. Born's rule is then P(aₙ) = |cₙ|², and completeness guarantees Σ|cₙ|² = ⟨ψ|ψ⟩ = 1, so the check costs nothing. Take ψ = (ψ₁ + √2 ψ₂ + ψ₃)/2 in the infinite well: the squared moduli are 1/4, 1/2 and 1/4, which do sum to 1, and ⟨E⟩ = (1/4)E₁ + (1/2)(4E₁) + (1/4)(9E₁) = 4.5E₁. Notice what that mean is not. It falls between E₂ = 4E₁ and E₃ = 9E₁ and equals neither, so no single electron ever registers it; it is the average of many readings on identically prepared electrons, and it agrees term by term with the integral ⟨ψ|Ĥ|ψ⟩. Notice also that only moduli entered. Put a phase on the ψ₂ term and the position density is rearranged — ψ₂ is antisymmetric about the midpoint, so even ⟨x⟩ slides off the centre of the well — while all three energy probabilities stay exactly where they were.

Born's rule gives the odds and says nothing about what you now hold. The projection postulate does: with P̂ₙ the projector onto the aₙ eigenspace, |ψ⟩ → P̂ₙ|ψ⟩ / √(⟨ψ|P̂ₙ|ψ⟩). That square root is nothing but √P(aₙ), so the division only restores unit norm — yet forgetting it is the commonest slip in the algebra. The reason to believe the rule at all is repeatability. Measure again immediately and P̂ₙ|aₙ⟩ = |aₙ⟩, so the same value returns with probability 1. Feed the +ħ/2 output of a Stern–Gerlach magnet into a second identical magnet and the − channel stays empty. When aₙ is degenerate the correct rule is Lüders': project onto the whole subspace and keep the relative phases inside it. The cruder "collapse picks one eigenvector" is a genuinely different state, and the Hard example below separates the two predictions by 0.90 against 0.50.

Position has no normalisable eigenstates — δ(x − x₀) has infinite norm and infinite kinetic energy — so the postulate cannot literally leave you in one. What a real detector implements is a projection onto an interval: ψ(x) → ψ(x)⋅χ(x) / ‖ψχ‖, where χ is 1 inside the accepted window and 0 outside. That state is normalisable, and probabilities come from ∫|ψ|²dx across the window rather than from |ψ|² at a point. It is also where the cost of a sharp reading shows up. Chop a wave function down to Δx = 1.0 nm and the momentum spread is at least ħ/(2Δx) = 1.055×10⁻³⁴ ÷ 2.0×10⁻⁹ = 5.3×10⁻²⁶ kg m s⁻¹, which for an electron is a velocity spread of 5.3×10⁻²⁶ ÷ 9.11×10⁻³¹ = 5.8×10⁴ m s⁻¹. The projection did not reveal a momentum the electron had; it rebuilt the momentum content by cutting the function.

Once an intermediate value has been recorded, probabilities multiply along the recorded path: P(aₙ then bₘ) = |⟨bₘ|aₙ⟩|² · |⟨aₙ|ψ⟩|². The word carrying the weight there is recorded. Prepare S_z = +ħ/2, send the beam through an Sₓ analyser, keep its + output and read S_z again: the Medium example counts the atoms, and the count is not zero, because the middle reading installed |+x⟩ and |+x⟩ holds no memory of z. Now rebuild the same hardware so the two x beams rejoin with nothing written down about which one an atom took. Not one magnet has moved, but the amplitudes now add before they are squared, they reassemble |↑z⟩ exactly, and the −z channel goes dark. Two apparatuses that look alike, two different answers: the postulate is applied where a value becomes a fact, and applying it one step too early predicts the wrong count.

Everything else in this unit runs on one linear equation, iħ ∂ψ/∂t = Ĥψ: deterministic, unitary, norm-preserving, and reversible. The update rule is none of those. It is discontinuous; it is non-unitary, since projection throws probability away before rescaling; and because of that rescaling it is not even linear in ψ. Applied to a superposition it yields one outcome, and a linear deterministic equation applied to the same superposition provably cannot. The two rules also disagree about jurisdiction. Nothing in the formalism marks the boundary between "system" and "apparatus", and you may slide that cut a long way — into the magnet, the detector, the experimenter — without changing any prediction anyone has tested. Decoherence explains part of the puzzle: coupling to an environment suppresses the interference terms extraordinarily fast for anything macroscopic. But it converts a superposition into a list of possibilities, not into one actuality. Use the postulate; it has never failed a measurement. Do not mistake it for a derivation.