Measurement, Collapse, and Spectral Expansion

Derivation

Measurement, Collapse, and Spectral Expansion

D-147 Home PU-202 Threads chance · symmetry Depends on Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates, Born Rule and Conservation of Normalisation

Statement

For an observable represented by a Hermitian operator Â with discrete, non-degenerate spectrum {a_n} and orthonormal eigenvectors {|a_n⟩}, any normalised state expands as |ψ⟩ = Σ_n c_n|a_n⟩ with c_n = ⟨a_n|ψ⟩. A measurement of Â then yields the value a_n with probability P(a_n) = |c_n|² = |⟨a_n|ψ⟩|², and immediately after such an outcome the system is left in the corresponding eigenstate, |ψ⟩ → |a_n⟩.

Why it matters

This is the bridge between the continuous, deterministic Schrödinger dynamics and the discrete, probabilistic facts we actually record in the laboratory. It tells you exactly which numbers can come out of an apparatus (the eigenvalues), how often each appears (the squared overlap), and what state to use for any subsequent calculation (the projected eigenstate).

Every practical prediction in quantum mechanics — spectral line intensities, Stern–Gerlach beam splittings, qubit readout statistics — is an application of this single expansion-and-projection rule. Without it, a state vector would be uninterpretable.

Assumptions

The observable is Hermitian with a discrete, complete, orthonormal eigenbasis.If the spectrum is not complete, |ψ⟩ has no unique expansion and the probabilities need not sum to one; if not orthonormal, the coefficients c_n are not clean overlaps and cross terms spoil the Born weights. The spectrum is non-degenerate.With degeneracy a single eigenvalue labels a whole subspace, so the outcome does not fix a unique post-measurement ray; collapse must be replaced by projection onto the eigenspace (Lüders rule). The Born rule and normalisation of |ψ⟩ are already established.These are the two imported prior results: without Born, |c_n|² is just a number, not a probability; without ⟨ψ|ψ⟩ = 1 the weights are unnormalised and must be divided through by ⟨ψ|ψ⟩. The measurement is ideal, projective, and instantaneous.If the coupling to the apparatus is weak, extended in time, or noisy, the sharp projection postulate is replaced by generalised (POVM / Kraus) updates and the clean eigenstate output is lost.

Derivation

Â|a_n⟩ = a_n|a_n⟩ , ⟨a_m|a_n⟩ = δ_mn

Spectral theorem for a Hermitian operator: real eigenvalues, and eigenvectors chosen orthonormal (imported result hermitian-operators-real-spectra). B

Σ_n |a_n⟩⟨a_n| = Î

Completeness (closure) of the eigenbasis: a complete orthonormal set resolves the identity. This is the structural input the whole derivation turns on. B

|ψ⟩ = Î|ψ⟩ = Σ_n |a_n⟩⟨a_n|ψ⟩ = Σ_n c_n|a_n⟩, c_n ≡ ⟨a_n|ψ⟩

Insert the resolution of identity from step 2 and read off the expansion coefficients as overlaps. This is pure algebra once closure is granted. A

1 = ⟨ψ|ψ⟩ = Σ_m,n c_m*c_n⟨a_m|a_n⟩ = Σ_n |c_n|²

Take the norm and use orthonormality (δ_mn) to collapse the double sum to a single one. Normalisation of |ψ⟩ is the imported result. A

P(a_n) = |c_n|² = |⟨a_n|ψ⟩|²

Apply the Born rule to the eigenbasis amplitudes: the probability of outcome a_n is the squared modulus of the corresponding coefficient. Step 4 guarantees these numbers form a valid probability distribution (non-negative, summing to one). B

|ψ⟩ ⟶_{outcome a_n} |a_n⟩ = P̂_n|ψ⟩ / |c_n| , P̂_n = |a_n⟩⟨a_n|

Projection postulate: on registering a_n, keep only the corresponding term of the expansion and renormalise. For a non-degenerate eigenvalue P̂_n is rank-one, so the result is the ray |a_n⟩ up to a phase. C

|⟨a_n|a_n⟩|² = 1 ⟹ repeat measurement returns a_n with certainty

Feed the post-measurement state back into step 5. Idempotence of the projector (P̂_n² = P̂_n) makes the rule self-consistent: an immediate re-measurement reproduces the outcome, matching the "reproducibility of sharp measurements" requirement. B

⟨Â⟩ = Σ_n a_n P(a_n) = Σ_n a_n|c_n|² = ⟨ψ|Â|ψ⟩

Form the statistical mean of the eigenvalues weighted by the probabilities of step 5, then run steps 2–3 in reverse to recognise it as the sandwich ⟨ψ|Â|ψ⟩. This ties the probabilistic reading to the operator expectation value with no extra assumption. C

Result

P(a_n) = |⟨a_n|ψ⟩|² and |ψ⟩ ⟶ |a_n⟩ on outcome a_n

Reading. Expand the state in the observable's eigenbasis; the modulus-squared of each coefficient is the probability of the associated eigenvalue, and the act of obtaining that eigenvalue discards every other term, leaving the matching eigenstate. The complete distribution {P(a_n)} carries everything Â can reveal about |ψ⟩; its mean is ⟨ψ|Â|ψ⟩.

Units check. The coefficients c_n = ⟨a_n|ψ⟩ are overlaps of unit-norm vectors, hence dimensionless; |c_n|² is therefore dimensionless, as a probability must be, and Σ|c_n|² = 1 is a pure number. The expectation value ⟨Â⟩ = Σ a_n|c_n|² inherits exactly the units of the eigenvalues a_n (e.g. joules for energy, ℏ for spin projection).

Limiting cases

State already an eigenstate (|ψ⟩ = |a_k⟩): c_n = δ_nk, so P(a_k) = 1 and the measurement is deterministic and non-disturbing.
Equal-weight two-state superposition (|ψ⟩ = (|a₁⟩ + e^iφ|a₂⟩)/√2): P(a₁) = P(a₂) = ½, independent of the phase φ.
Large-N uniform spread (|c_n|² = 1/N): each of N outcomes is equally likely; the distribution flattens toward maximal ignorance about the result.
Degenerate limit: the rank-one projector of step 6 widens to P̂_n = Σ_k|a_n,k⟩⟨a_n,k|, and collapse lands in a subspace rather than on a single ray.

Breaks when

Continuous spectrum. For position or momentum the sum becomes an integral, |c_n|² becomes a probability density |⟨x|ψ⟩|² (units of inverse length), and the eigenstates are non-normalisable (δ-normalised); a single sharp outcome has zero probability, so one must speak of intervals and the collapse is to a narrow packet, not a delta.
Degenerate eigenvalues. The naive rule |ψ⟩ → |a_n⟩ is undefined because the eigenvalue no longer names a unique vector; probabilities require the subspace projector, P(a_n) = ⟨ψ|P̂_n|ψ⟩, and collapse follows the Lüders rule |ψ⟩ → P̂_n|ψ⟩/√⟨ψ|P̂_n|ψ⟩.
Non-ideal measurement. Weak, noisy, or finite-duration couplings violate the projective idealisation; the update is then a Kraus map Σ_k M̂_kρM̂_k† and outcomes are POVM elements, not eigenprojectors — states no longer land cleanly on eigenvectors.
Dynamics during readout. If [Ĥ, Â] ≠ 0 and the measurement is not instantaneous, unitary evolution competes with projection; the post-measurement eigenstate immediately starts to spread, and repeated "sharp" measurements no longer return identical values.

Failure modes

Amplitude–probability confusion: quoting c_n itself (or Re c_n) as the probability instead of |c_n|². Only the squared modulus is physical.
Skipping normalisation: reading |c_n|² off an unnormalised state, so the "probabilities" do not sum to one; you must divide by ⟨ψ|ψ⟩ first.
Adding amplitudes across outcomes: treating distinguishable eigenvalues like interfering paths. Distinct measurement outcomes add in probability, |c₁|² + |c₂|², never in amplitude.
Collapsing a degenerate outcome to one ket: writing |ψ⟩ → |a_n⟩ when a_n is degenerate, silently discarding the surviving superposition inside the eigenspace.
Global-phase or basis error: expanding in the wrong (non-eigen) basis of Â, or believing the overall phase of |ψ⟩ affects P(a_n). Probabilities depend only on |ψ⟩ as a ray, in Â's own eigenbasis.

Discussion

The expansion |ψ⟩ = Σ_n c_n|a_n⟩ is not a physical process but a change of description: the same state written in the basis adapted to the observable we intend to measure. Completeness guarantees the description loses nothing, and orthonormality guarantees the coefficients are independent handles. Measurement is where the linear, reversible algebra gives way to a single, irreversible selection — the one genuinely non-unitary step in the theory.

Because P(a_n) = |⟨a_n|ψ⟩|² depends only on the overlap, it is invariant under a global phase of |ψ⟩ and under any phase attached to |a_n⟩. This is why physical states are rays rather than vectors, and why the relative phases between coefficients — invisible to a measurement of Â — nonetheless govern the statistics of a complementary observable B̂ that does not commute with Â. The chosen measurement basis decides which phase information is exposed and which is hidden.

The projection of step 6 is what makes quantum information storable and reproducible: after a sharp measurement the state is pinned to a known eigenstate, so an immediate repeat confirms it. Iterated on a slowly evolving system this yields the quantum Zeno effect, where frequent measurement freezes the dynamics. It also underlies qubit readout, where a projective measurement both extracts a bit and prepares the register in a definite computational-basis state.

Whether step 6 describes a real physical event or merely an updating of our information is the substance of the measurement problem. Unitary Schrödinger evolution alone never produces the single outcome; the projection is grafted on as a separate postulate. Decoherence explains why the interference between eigenstates becomes practically unobservable once the apparatus and environment entangle with the system — it selects the pointer basis and diagonalises the reduced density matrix — but it does not by itself pick the one outcome that occurs. Interpretations (Copenhagen, many-worlds, objective-collapse, Bohmian) differ precisely in what they say happens at this line, while agreeing on every |c_n|² this derivation produces.

Common misconceptions. Collapse is not a signal that propagates or a force that acts; nothing travels. The state vector is a catalogue of probabilities, and "collapse" is the abrupt replacement of that catalogue when new information (the outcome) is acquired. Likewise, superposition is not "the system being in several states at once" in any classical sense — it is a single state whose measurement statistics in a given basis are governed by the squared overlaps derived here.

Worked examples

Example 1 — Spin-½ measured along z.

|ψ⟩ = cos(θ/2)|↑⟩ + e^iφ sin(θ/2)|↓⟩, Ŝ_z|↑⟩ = +(ℏ/2)|↑⟩, Ŝ_z|↓⟩ = −(ℏ/2)|↓⟩

Bloch-sphere state; {|↑⟩,|↓⟩} is the orthonormal eigenbasis of Ŝ_z. Symbols first. A

c_↑ = ⟨↑|ψ⟩ = cos(θ/2), P(+ℏ/2) = cos²(θ/2)

Overlap picks the coefficient (step 3 of the derivation), Born rule squares it (step 5). B

θ = 60°: P(+ℏ/2) = cos²30° = 0.750, P(−ℏ/2) = sin²30° = 0.250

Numbers substituted; probabilities sum to 1.000, and note the phase φ dropped out entirely. A

⟨Ŝ_z⟩ = (+ℏ/2)(0.750) + (−ℏ/2)(0.250) = (ℏ/2)(0.500) = ℏ/4 = (ℏ/2)cosθ

Mean of eigenvalues weighted by probabilities (step 8); matches the standard (ℏ/2)cosθ. B

P(↑) = 0.750, P(↓) = 0.250, ⟨Ŝ_z⟩ = ℏ/4; on "↑" the spin collapses to |↑⟩

Reading. A spin tilted 60° from z reads "up" three times out of four; each such readout leaves it in |↑⟩, so an immediate re-measurement is certain to give +ℏ/2. Units: probabilities dimensionless; ⟨Ŝ_z⟩ carries units of ℏ (angular momentum), as required.

Example 2 — Energy measured on a three-level superposition.

|ψ⟩ = (2|E₁⟩ + |E₂⟩ − |E₃⟩)/√6, E₁ = 1.0 eV, E₂ = 2.0 eV, E₃ = 3.0 eV

Energy eigenstates orthonormal; Ĥ|E_n⟩ = E_n|E_n⟩. Check norm: (2²+1²+1²)/6 = 6/6 = 1. A

P(E₁) = |2/√6|² = 4/6 = 2/3, P(E₂) = 1/6, P(E₃) = 1/6

Born rule on each coefficient (step 5); Σ = 2/3 + 1/6 + 1/6 = 1. B

⟨Ĥ⟩ = Σ E_nP(E_n) = (1.0)(2/3) + (2.0)(1/6) + (3.0)(1/6) eV

Expectation value from step 8, symbols then numbers. B

⟨Ĥ⟩ = (4/6 + 2/6 + 3/6)·(... ) = 9/6 eV = 1.5 eV; outcome E₂ ⟹ |ψ⟩ → |E₂⟩

Arithmetic complete; on registering 2.0 eV the state collapses to |E₂⟩ (step 6). A

P = {2/3, 1/6, 1/6}, ⟨Ĥ⟩ = 1.5 eV, post-measurement state = |E₂⟩ if 2.0 eV read

Reading. The ground level dominates the statistics (two-thirds of runs); the mean energy 1.5 eV lies between the levels and is not itself an allowed outcome. Units: weights dimensionless, ⟨Ĥ⟩ in eV (energy), matching the eigenvalues.

Problems

A qubit is in |ψ⟩ = (√3 |0⟩ + i|1⟩)/2. Measuring in the computational basis, find P(0), P(1), and the state just after obtaining "1".
Solution
c₀ = √3/2, c₁ = i/2. P(0) = |√3/2|² = 3/4 = 0.75; P(1) = |i/2|² = 1/4 = 0.25 (the phase i is irrelevant to the modulus). Sum = 1. On outcome "1" the state collapses to |1⟩ (the phase i is a global phase of the surviving ray and carries no physical content).
An unnormalised state is written |χ⟩ = 3|a₁⟩ − 4|a₂⟩ in the orthonormal eigenbasis of Â (eigenvalues a₁ = +5, a₂ = −5). Find the measurement probabilities and ⟨Â⟩.
Solution
Norm² = 3² + 4² = 25, so normalise by dividing coefficients by 5: c₁ = 3/5, c₂ = −4/5. P(a₁) = 9/25 = 0.36; P(a₂) = 16/25 = 0.64; sum = 1. ⟨Â⟩ = (+5)(0.36) + (−5)(0.64) = 1.8 − 3.2 = −1.4 (same units as the eigenvalues).
For the state of Example 2, compute the variance (ΔĤ)² = ⟨Ĥ²⟩ − ⟨Ĥ⟩² and the standard deviation ΔĤ.
Solution
⟨Ĥ²⟩ = Σ E_n²P(E_n) = (1.0²)(2/3) + (2.0²)(1/6) + (3.0²)(1/6) = 2/3 + 4/6 + 9/6 = (4 + 4 + 9)/6 = 17/6 ≈ 2.833 eV². ⟨Ĥ⟩² = (1.5)² = 2.25 eV². (ΔĤ)² = 2.833 − 2.25 = 0.583 eV², so ΔĤ = √0.583 ≈ 0.76 eV.
An observable Â has a degenerate eigenvalue a = 2 spanned by {|u⟩, |v⟩} and a non-degenerate eigenvalue b = 5 with eigenvector |w⟩. The state is |ψ⟩ = (|u⟩ + |v⟩ + |w⟩)/√3. Find P(2), and the (normalised) state just after obtaining the value 2.
Solution
Projector onto the a = 2 eigenspace: P̂ = |u⟩⟨u| + |v⟩⟨v|. P̂|ψ⟩ = (|u⟩ + |v⟩)/√3, with norm² = (1 + 1)/3 = 2/3. So P(2) = ⟨ψ|P̂|ψ⟩ = 2/3. By the Lüders rule the post-measurement state is P̂|ψ⟩/√(2/3) = (|u⟩ + |v⟩)/√2 — a surviving superposition inside the eigenspace, not a single eigenvector.
A spin-½ starts in |↑_z⟩. It is measured along x (eigenstates |±_x⟩ = (|↑_z⟩ ± |↓_z⟩)/√2), and then measured again along z. Find the probability that the final z-measurement gives "down".
Solution
First measurement: P(+_x) = |⟨+_x|↑_z⟩|² = |1/√2|² = 1/2, and likewise P(−_x) = 1/2. Whichever x-outcome occurs, the state collapses to |±_x⟩. Second measurement: from |+_x⟩, P(↓_z) = |⟨↓_z|+_x⟩|² = |1/√2|² = 1/2; from |−_x⟩, P(↓_z) = |⟨↓_z|−_x⟩|² = |−1/√2|² = 1/2. Total: P(↓_z) = (1/2)(1/2) + (1/2)(1/2) = 1/2. The intermediate x-measurement erased the original certainty of "up", illustrating how collapse in one basis randomises a non-commuting observable.