3.8 Measurement and the projection postulate

The previous section laid the measurement postulate on the table. This section takes it apart.

What the postulate says

Let the observable $\hat A$ have the spectral decomposition $\hat A=\sum_n a_n\hat P_n$ , where $\hat P_n$ projects onto the eigenspace of the eigenvalue $a_n$ .

The picture

Two things, happening at once

(1) Outcome and probability

P(a_n)=\bra{\psi}\hat P_n\ket{\psi}=\big\|\hat P_n\ket{\psi}\big\|^2\tag{3.8.1}

In the non-degenerate case $\hat P_n=\ket{a_n}\bra{a_n}$ , and this reduces to the familiar $P=|\braket{a_n}{\psi}|^2$ .

(2) The post-measurement state

\ket{\psi'}=\frac{\hat P_n\ket{\psi}}{\big\|\hat P_n\ket{\psi}\big\|}\tag{3.8.2}

The denominator is just renormalisation.

The mathematics

The probabilities sum to 1 automatically

\sum_n P(a_n)=\bra{\psi}\Big(\sum_n\hat P_n\Big)\ket{\psi}=\braket{\psi}{\psi}=1\tag{3.8.3}

using completeness, $\sum_n\hat P_n=\hat 1$ .

No coincidence: it is a direct consequence of the spectral theorem of section 3.5 — probability normalisation comes free with “observables must be Hermitian”.

See it with your own hands

The infinite square well

Drag the width L and the quantum number n and watch the energy, the waveform and the probability distribution move together. Units: ħ = m = 1.

Energy Eₙ (n=1): 4.935
relative to E₁(L=1): 1.00 ×
⟨x⟩ / L: 0.500
Δx / L: 0.180

Well width L1.00

Shrink the box and every level is pushed up together: E ∝ 1/L²

Quantum number n1

nodes = n − 1 = 0 (endpoints excluded)

State

Show ψShow |ψ|²

Try this

Drag L from 2.5 down to 0.6 and keep your eye on the dashed line in the level diagram (pinned at the reference energy E₁(L=1)): it sinks all the way to the bottom, meaning every level has risen far above it. The "relative to E₁(L=1)" readout climbs from 0.16 to 2.78 — "the tighter you confine a particle, the more kinetic energy it has", a direct consequence of ΔxΔp ≥ ħ/2.
Change n in the stationary mode and note that |ψ|² never changes with time (the curve stands still); switch to a superposition, press play, and |ψ|² immediately starts sloshing. That is exactly what "stationary" refers to.
Leave only c₁ in the superposition (drag the rest to 0) and press play — the probability density stops moving again. However long a single eigenstate evolves, it only picks up an overall phase e^(−iEₙt/ħ), which no observable can see.
Push n above 10 and look at |ψ|²: the fringes get so fine that the distribution is nearly uniform — the classical picture of a particle equally likely to be anywhere in the box. The correspondence principle, in view.

This is an interactive simulation (infinite-well) and needs JavaScript to run. The derivations and conclusions in the text do not depend on it, so feel free to read on.

Switch the mode to “superposition” and make both $c_1$ and $c_2$ non-zero. By postulate 3, measuring the energy now:

gives $E_1$ with probability $|c_1|^2$ and $E_2$ with probability $|c_2|^2$ ;
once $E_1$ is obtained, the state instantly becomes pure $\psi_1$ — the probability density stops sloshing.

That is exactly what “dragging $c_2$ to 0” does in the simulation. The difference: you drag by hand, whereas the measurement happens at random — but once it happens, the outcome is the same.

What the postulate does not say

This may be the most important part of the section.

Questions the projection postulate leaves unanswered

1. At which step does the collapse happen?

The postulate only says “after the measurement” — without defining what counts as a measurement. When the particle hits the detector? When the current is amplified? When the data are written to disk? When somebody looks at the screen?

On this the postulate is entirely silent.

2. How long does the collapse take?

The postulate says “immediately”. But real measurement processes take time. The theory gives no dynamics of collapse — because it is not a dynamical process at all; it is a substitution written into a postulate.

3. Why this outcome and not that one?

The postulate gives only probabilities. As for why this particular run gave $E_1$ , the theory says: there is no reason.

This is fundamental randomness, not “we just don’t know”. Bell experiments (section 9.4) have ruled out, at extremely high confidence, local hidden-variable stories of the kind “the answer was there all along, we simply had not looked”.

Three widely circulated wrong claims

“Consciousness causes collapse.” Wigner once took this seriously, and later abandoned it himself. It has no experimental support whatsoever, and it faces an awkward question: before conscious observers existed, did any collapse ever happen in the universe? Mainstream physics does not adopt this view.

“Measurement disturbs because the detector gives the particle a knock.” Sometimes true, but not the essence. The quarter-wave-plate scheme of section 1.6 exchanges no momentum, yet still wipes out the interference. What does the work is the recording of information, not the exchange of momentum.

“Collapse is faster than light, so it violates relativity.” The collapse of an entangled state is indeed “nonlocal”, but it cannot be used to send signals. Alice’s outcomes look completely random to Bob; the statistics in his hands do not change no matter what Alice does. Only by comparing records over a classical channel can the correlations be seen — and classical channels are not faster than light. This is the no-communication theorem, worked out in section 9.5.

The precise location of the measurement problem

The picture

Treat the apparatus as a quantum system too

Let the measured system start in $\ket{\psi}=c_1\ket{a_1}+c_2\ket{a_2}$ and the apparatus in $\ket{M_0}$ .

By postulate 4 (unitary evolution), a suitable interaction gives

\ket{\psi}\ket{M_0}\longrightarrow c_1\ket{a_1}\ket{M_1}+c_2\ket{a_2}\ket{M_2}\tag{3.8.5}

The apparatus has faithfully recorded the result — but what it recorded is a superposition.

The Schrödinger equation delivers “a pointer aiming at two readings at once”. And nobody has ever seen such a thing.

The mathematics

Where the seam is

\underbrace{c_1\ket{a_1}\ket{M_1}+c_2\ket{a_2}\ket{M_2}}_{\text{what postulate 4 gives}}\tag{3.8.6}

\Big\downarrow\ ?\tag{3.8.7}

\underbrace{\ket{a_1}\ket{M_1}\ \text{or}\ \ket{a_2}\ket{M_2}}_{\text{what we see}}\tag{3.8.8}

This step cannot come from postulate 4: unitary evolution is linear, it turns superpositions into superpositions, and it never singles out one term.

Schrödinger’s cat is the same argument with the apparatus swapped for a cat.

What decoherence achieves — and what it does not

The apparatus is not isolated: it interacts ceaselessly with its environment (air molecules, thermal radiation, internal degrees of freedom). Write the environment in too:

c_1\ket{a_1}\ket{M_1}\ket{E_1}+c_2\ket{a_2}\ket{M_2}\ket{E_2}\tag{3.8.9}

For a macroscopic apparatus the two environment states become orthogonal almost instantly, $\braket{E_1}{E_2}\approx0$ . Trace out the environment and the off-diagonal elements of the apparatus’s reduced density matrix are exponentially suppressed:

\rho_M\to\begin{pmatrix}|c_1|^2&\approx0\\ \approx0&|c_2|^2\end{pmatrix}\tag{3.8.10}

The timescale is absurdly short: for a dust grain in air, about $10^{-31}$ seconds.

What decoherence solves: why macroscopic objects never show interference between superposed states; why “pointer position”, and not some other basis, gets singled out (the preferred-basis problem); why collapse looks instantaneous.

What decoherence does not solve: the matrix above is a mixed state, read as “it is 1 or 2, we just don’t know which”. But mathematically it is still part of the full entangled state — the global state remains a superposition. Why we experience only one outcome, decoherence does not answer.

This is worth stating plainly, because “decoherence solved the measurement problem” is a widespread oversimplification. It shrank the problem from “why no macroscopic superpositions” to “why a single outcome” — enormous progress, but not the end. Section 10.5 carries out the full calculation.

What each interpretation gives up

Interpretation	What it gives up	The price
Copenhagen	asking about the collapse mechanism	“measurement” remains forever a primitive
Many-worlds	postulate 3 (a unique outcome)	all branches actually exist; the origin of probability needs a separate argument
Bohmian mechanics	locality	explicitly nonlocal, sitting awkwardly with relativity
GRW / spontaneous collapse	strictly unitary evolution	modifies the Schrödinger equation; falsifiable in principle
Relational / QBism	objectivity of the state	the state becomes “observer-relative” or “a belief”

Look at the last column. These are not “matters of taste” — each pays a genuine price somewhere, and the GRW family actually makes predictions that differ from standard quantum mechanics, with experiments steadily tightening the bounds.

All interpretations give identical predictions for every experiment so far, so this is not a problem that more precise calculation can settle. See section 13.7.

Key formulas

Measurement probability (with degeneracy)
$P(a_n)=\bra{\psi}\hat P_n\ket{\psi}=\|\hat P_n\ket{\psi}\|^2$
Reduces to |⟨aₙ|ψ⟩|² when non-degenerate
Post-measurement state
$\ket{\psi'}=\hat P_n\ket{\psi}/\|\hat P_n\ket{\psi}\|$
The denominator is just renormalisation
Probability normalisation
$\sum_n P(a_n)=\bra{\psi}\hat 1\ket{\psi}=1$
A free gift of the spectral theorem
Repeatability
$P(a_n\ \text{again})=1$
Persists indefinitely only when [Â,Ĥ]=0
Location of the measurement problem
$c_1\ket{a_1}\ket{M_1}+c_2\ket{a_2}\ket{M_2}\ \not\to\ \text{a single term}$
Linear evolution cannot single out one term

Self-check4 questions

What comes next

Measurement is the one non-unitary step in the theory. The rest of the time, states evolve by postulate 4.

The next section packages that evolution into a single operator and introduces something very practical: whether the time dependence sits on the states or on the operators is a free choice.

Measurement and the projection postulate

Recommended first

After this section you should be able to

What the postulate says

The picture

The mathematics

See it with your own hands

The infinite square well

What the postulate does not say

The precise location of the measurement problem

The picture

The mathematics

Key formulas

Self-check4 questions

What comes next

What the postulate says

◑The picture

∑The mathematics

See it with your own hands

What the postulate does not say

The precise location of the measurement problem

◑The picture

∑The mathematics

∎Key formulas

What comes next

The picture

The mathematics

The picture

The mathematics

Key formulas