---
title: The Postulates and Quantum Measurement
module: The Formalism of Quantum Mechanics
moduleNumber: 4
lessonNumber: 3
order: 403
summary: >
  With states as vectors and observables as Hermitian operators, the physical
  content of quantum mechanics reduces to a short list of postulates. We state them
  precisely, derive the Born probability rule for discrete and continuous spectra,
  work out projective collapse and its idempotence, compute expectation values and
  their variance, and state the measurement problem cleanly — the one place the
  postulates split unitary evolution from measurement without explaining the seam.
topics: [The Formalism of Quantum Mechanics]
sources:
  - book: Sakurai & Napolitano
    ref: "Ch. 1; §1.4 Measurements, Observables, and the Uncertainty Relations"
  - book: Griffiths & Schroeter
    ref: "Ch. 3; §3.4 The Generalized Statistical Interpretation; Ch. 4 §4.5"
  - book: Cohen-Tannoudji, Diu & Laloë
    ref: "Ch. III — The Postulates of Quantum Mechanics"
draft: false
---

The formalism so far is mathematics: a
[Hilbert space](/quantum-mechanics/formalism/hilbert-space-and-dirac-notation) of
states and [Hermitian operators](/quantum-mechanics/formalism/observables-hermitian-operators-and-eigenvalues)
that act on them. Physics enters through a short list of postulates that connect
the mathematics to laboratory outcomes: which vector describes a system, which
operator represents a quantity, what numbers a measurement can return, with what
probability, and how the state responds to being measured. Every quantitative
prediction in the subject is an application of these rules.

## The postulates

The standard formulation collects into five statements. They are stated here in the
basis-free language of the preceding lessons; the position-representation versions
used in wave mechanics are special cases.

> **Postulate I (State).** At a fixed time the state of a system is a normalized
> vector $\ket{\psi}$ in a Hilbert space $\mathcal{H}$, with
> $\braket{\psi|\psi}=1$. Vectors differing only by an overall phase
> $e^{i\theta}\ket{\psi}$ describe the same physical state.

> **Postulate II (Observables).** Each measurable quantity $A$ is represented by a
> Hermitian operator $\hat A$ on $\mathcal{H}$. The possible outcomes of a
> measurement of $A$ are the eigenvalues $a_n$ of $\hat A$, all real.

> **Postulate III (Born rule).** If the system is in state $\ket{\psi}$, a
> measurement of $A$ yields the eigenvalue $a_n$ with probability
> $$P(a_n) = \lVert \hat P_n\ket{\psi}\rVert^2 = \braket{\psi|\hat P_n|\psi},$$
> where $\hat P_n$ projects onto the eigenspace of $a_n$. For a nondegenerate
> eigenvalue this is $P(a_n)=\lvert\braket{a_n|\psi}\rvert^2$.

> **Postulate IV (Collapse).** Immediately after a measurement returning $a_n$, the
> state is the normalized projection onto that eigenspace,
> $$\ket{\psi} \;\longmapsto\; \frac{\hat P_n\ket{\psi}}{\lVert \hat P_n\ket{\psi}\rVert}.$$

> **Postulate V (Time evolution).** Between measurements the state evolves
> deterministically by the Schrödinger equation
> $i\hbar\,\partial_t\ket{\psi}=\hat H\ket{\psi}$, developed in the lesson on
> [time evolution](/quantum-mechanics/formalism/time-evolution-schrodinger-and-heisenberg-pictures).

Postulates III and IV are the whole of measurement. The remaining lessons of the
module draw out their consequences; this one derives the machinery they imply.

## The Born rule and probability

The Born rule is the bridge from amplitudes to frequencies. Expand a normalized
state in the eigenbasis of $\hat A$ (taking the spectrum nondegenerate for
clarity),

$$
\ket{\psi} = \sum_n c_n\ket{a_n}, \qquad c_n = \braket{a_n|\psi},
$$

so the probability of outcome $a_n$ is $P(a_n)=\lvert c_n\rvert^2$. Two facts make
this a consistent probability assignment, both consequences of the earlier
formalism.

- **Normalization to one.** The probabilities sum to unity because the eigenbasis
  is complete:
  $$
  \sum_n P(a_n) = \sum_n \lvert c_n\rvert^2 = \sum_n\braket{\psi|a_n}\braket{a_n|\psi}
  = \braket{\psi|\Bigl(\sum_n\ket{a_n}\bra{a_n}\Bigr)|\psi} = \braket{\psi|\psi} = 1.
  $$
  Normalization of the probabilities follows directly from the
  [resolution of the identity](/quantum-mechanics/formalism/hilbert-space-and-dirac-notation).
- **Basis independence.** $P(a_n)$ depends only on the state and the operator, not
  on any auxiliary basis, because $\hat P_n$ is defined by $\hat A$ alone.

The projection form $P(a_n)=\braket{\psi|\hat P_n|\psi}$ is the general statement,
correct for degenerate spectra where several orthonormal eigenvectors share $a_n$;
then $\hat P_n=\sum_k\ket{a_n,k}\bra{a_n,k}$ and
$P(a_n)=\sum_k\lvert\braket{a_n,k|\psi}\rvert^2$ sums the weights over the whole
eigenspace.

The overall phase of the state drops out of every probability, which is why
Postulate I identifies $\ket{\psi}$ and $e^{i\theta}\ket{\psi}$: multiplying the
state by a global phase multiplies each amplitude $c_n$ by the same $e^{i\theta}$,
leaving $\lvert c_n\rvert^2$ untouched. Relative phases between components, by
contrast, are fully physical — they determine the outcome statistics of any
observable that does not commute with $\hat A$. The distinction between an
unobservable global phase and observable relative phases is the reason the physical
state space is the projective space of rays, not the Hilbert space of vectors, a
point that returns in the [geometry of two-level systems](/quantum-mechanics/spin/two-level-systems-and-the-bloch-sphere).

$$
% caption: The Born rule reads a superposition as a probability distribution over
% outcomes: each eigenstate's squared amplitude is the height of a bar, and the
% bars sum to one by completeness of the eigenbasis.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (6.4,0) node[right, black!70] {outcome $a_n$};
\draw[black, ->] (0,0) -- (0,3.1) node[above, black!70] {$P(a_n)$};
\foreach \x/\h/\lab in {0.8/1.1/{a_1}, 2.0/2.6/{a_2}, 3.2/1.7/{a_3}, 4.4/0.7/{a_4}, 5.6/0.4/{a_5}}{
  \draw[acc, very thick] (\x,0) -- (\x,\h);
  \fill[acc] (\x,\h) circle (1.8pt);
  \node[anchor=north, black!70] at (\x,-0.08) {$\lab$};
}
\node[acc, anchor=south] at (2.0,2.6) {peak weight};
\end{tikzpicture}
$$

### Continuous spectra

For an observable with continuous spectrum — position is the standard case — the
eigenvalues form a continuum and the probability of an exact value is zero. The
Born rule becomes a probability **density**. With position eigenkets normalized to
$\braket{x|x'}=\delta(x-x')$ and $\psi(x)=\braket{x|\psi}$, the completeness relation
is an integral, and

$$
1 = \braket{\psi|\psi} = \int\braket{\psi|x}\braket{x|\psi}\d x
= \int \lvert\psi(x)\rvert^2\d x.
$$

The quantity $\lvert\psi(x)\rvert^2$ is therefore the probability density: the
probability of finding the particle in $[x,x+\d x]$ is $\lvert\psi(x)\rvert^2\d x$,
and the probability of landing in a region $\Omega$ is $\int_\Omega\lvert\psi\rvert^2\d x$.
This is Born's rule for continuous spectra, and it recovers the interpretation
introduced in [wave mechanics](/quantum-mechanics/wave-mechanics-1d/the-schrodinger-equation-in-one-dimension)
as a special case of the general postulate. The delta-normalized eigenkets and the
subtleties they carry are the subject of the
[continuous-spectra lesson](/quantum-mechanics/formalism/position-momentum-and-continuous-spectra).

$$
% caption: For a continuous spectrum the Born rule gives a probability density: the
% probability of an outcome in a window is the area under the density over that
% window, and the total area is one.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (6.4,0) node[right, black!70] {$x$};
\draw[black, ->] (0,0) -- (0,3.0) node[above, black!70] {density};
\draw[acc, very thick] plot[domain=0.2:6.1, samples=140]
  (\x, {2.3*exp(-1.1*(\x-2.6)^2) + 1.3*exp(-1.6*(\x-4.4)^2)});
% shaded window
\begin{scope}
\clip (3.4,0) rectangle (4.6,3.0);
\fill[acc!16] plot[domain=0.2:6.1, samples=140]
  (\x, {2.3*exp(-1.1*(\x-2.6)^2) + 1.3*exp(-1.6*(\x-4.4)^2)}) -- (6.1,0) -- (0.2,0) -- cycle;
\end{scope}
\draw[black] (3.4,0) -- (3.4,2.2);
\draw[black] (4.6,0) -- (4.6,2.2);
\node[anchor=north, acc] at (4.0,-0.1) {window};
\end{tikzpicture}
$$

## Expectation values

Repeating a measurement on many identically prepared copies and averaging the
outcomes gives the **expectation value**, the mean of the eigenvalues weighted by
the Born probabilities,

$$
\langle \hat A\rangle = \sum_n a_n P(a_n) = \sum_n a_n\braket{\psi|\hat P_n|\psi}
= \braket{\psi|\Bigl(\sum_n a_n\hat P_n\Bigr)|\psi} = \braket{\psi|\hat A|\psi},
$$

where the spectral decomposition $\hat A=\sum_n a_n\hat P_n$ collapses the weighted
sum into a single sandwich. The compact result

$$
\;\langle \hat A\rangle = \braket{\psi|\hat A|\psi}\;
$$

holds for any observable and any state, and it is the working definition of
expectation value throughout the subject. The **variance** measures the spread of
outcomes,

$$
\sigma_A^2 = \langle \hat A^2\rangle - \langle \hat A\rangle^2
= \braket{\psi|(\hat A-\langle \hat A\rangle)^2|\psi} \ge 0,
$$

and vanishes precisely when $\ket{\psi}$ is an eigenstate of $\hat A$, the only
case in which the observable has a **determinate** value. The variance is the
quantity bounded below by the
[uncertainty principle](/quantum-mechanics/formalism/commutators-and-the-generalized-uncertainty-principle)
for incompatible observables.

> **Worked example.** A spin-$\tfrac12$ system is prepared in
> $\ket{\psi}=\cos\tfrac\theta2\ket{\uparrow}+\sin\tfrac\theta2\ket{\downarrow}$,
> and $S_z=\tfrac\hbar2\sigma_z$ is measured. The outcomes are
> $\pm\tfrac\hbar2$ with probabilities
> $$
> P\!\left(+\tfrac\hbar2\right)=\cos^2\tfrac\theta2,
> \qquad P\!\left(-\tfrac\hbar2\right)=\sin^2\tfrac\theta2,
> $$
> which sum to one. The expectation value is
> $$
> \langle S_z\rangle = \tfrac\hbar2\cos^2\tfrac\theta2 - \tfrac\hbar2\sin^2\tfrac\theta2
> = \tfrac\hbar2\cos\theta,
> $$
> the projection of a classical spin of length $\tfrac\hbar2$ onto the $z$-axis. The
> variance follows from $S_z^2=\tfrac{\hbar^2}{4}\mathbb{1}$, so
> $\langle S_z^2\rangle=\tfrac{\hbar^2}{4}$ and
> $$
> \sigma_{S_z}^2 = \tfrac{\hbar^2}{4} - \tfrac{\hbar^2}{4}\cos^2\theta
> = \tfrac{\hbar^2}{4}\sin^2\theta,
> $$
> maximal at $\theta=\tfrac\pi2$ (equal superposition, most uncertain) and zero at
> $\theta=0$ (the eigenstate $\ket{\uparrow}$).

## Projective collapse

Postulate IV replaces the state by its normalized projection onto the measured
eigenspace. The projector structure makes the rule internally consistent under
immediate repetition.

> **Theorem (Repeatability).** A measurement of $\hat A$ that returned $a_n$, if
> repeated immediately, returns $a_n$ again with certainty.

> **Proof.** After the first measurement the state is
> $\ket{\psi'}=\hat P_n\ket{\psi}/\lVert\hat P_n\ket{\psi}\rVert$, which lies in the
> eigenspace of $a_n$, so $\hat P_m\ket{\psi'}=\delta_{mn}\ket{\psi'}$ by
> orthogonality of the projectors. The Born probability of outcome $a_m$ on the
> repeated measurement is
> $\braket{\psi'|\hat P_m|\psi'}=\delta_{mn}$: outcome $a_n$ has probability one and
> every other outcome probability zero.

The mathematical fact behind repeatability is **idempotence**, $\hat P_n^2=\hat P_n$:
projecting a state that is already in the eigenspace does nothing. A projective
measurement is a filter that either passes a state (if it already lies in the
eigenspace) or forces it into the eigenspace, and re-filtering has no further
effect.

The collapse rule above is the **selective** measurement — the state conditioned on
a specific outcome. If instead the measurement is performed but the result is not
recorded, the appropriate description is the statistical mixture of all possible
post-measurement states weighted by their probabilities. This **non-selective**
measurement takes a pure state to

$$
\ket{\psi}\bra{\psi} \;\longmapsto\; \sum_n \hat P_n\ket{\psi}\bra{\psi}\hat P_n,
$$

which erases the off-diagonal terms of the state in the eigenbasis of $\hat A$ while
leaving the diagonal probabilities $P(a_n)$ intact. That loss of coherence, phrased
in the language of the density operator, is the formal content of measurement-induced
decoherence.
Compatible measurements can be interleaved without conflict: if $[\hat A,\hat B]=0$,
measuring $\hat A$ then $\hat B$ then $\hat A$ returns the same $\hat A$ value both
times, because the intermediate $\hat B$ measurement collapses only within the
shared eigenbasis and never disturbs the $\hat A$ eigenvalue.

$$
% caption: Projective measurement collapses a state onto the measured eigenspace
% and normalizes it; a repeated measurement finds the state already inside and
% returns the same value with certainty (idempotence of the projector).
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, ->] (0,0) -- (4.4,0) node[right, black!70] {$a_1$ axis};
\draw[black, ->] (0,0) -- (0,3.4) node[above, black!70] {$a_2$ axis};
\coordinate (V) at (3.4,2.6);
\draw[->, black, very thick] (0,0) -- (V) node[above right, black!70] {before};
\draw[->, acc, very thick] (0,0) -- (0,2.6) node[midway, left, acc] {after};
\draw[black, dashed] (V) -- (0,2.6);
\node[anchor=north, black!70] at (2.2,-0.2) {measure A, get $a_2$};
\end{tikzpicture}
$$

The two dynamical laws of the theory now sit side by side and are visibly
different: Postulate V is deterministic, continuous, and unitary; Postulate IV is
probabilistic, discontinuous, and non-unitary. Nothing in the formalism specifies
when one takes over from the other.

## The measurement problem

The postulates work — they predict every observed statistic — but they contain a
seam. Two evolution laws govern the state: unitary Schrödinger flow between
measurements, and non-unitary projective collapse at a measurement. The postulates
do not say what physically constitutes a "measurement," where the boundary between
system and apparatus lies, or why a superposition of pointer states is never
observed. This is the **measurement problem**.

> **Definition (Measurement problem).** The formalism supplies two incompatible
> evolution laws — deterministic unitary evolution and stochastic collapse — and no
> internal criterion for when the second applies. Since an apparatus is itself a
> quantum system, unitary evolution alone would carry a superposed input into a
> superposition of macroscopically distinct pointer readings, which is never seen.

The problem is not that the predictions are wrong or ambiguous — for any concrete
measurement the rules give unambiguous probabilities. The difficulty is
conceptual: the theory divides the world into a quantum system and a classical
measuring device without saying where the cut belongs.

$$
% caption: The measurement seam: unitary evolution alone maps a superposed system
% plus ready apparatus to an entangled superposition of pointer readings, whereas
% experience shows a single definite outcome; the postulates insert collapse by
% hand at the cut.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0,
  box/.style={draw, minimum width=30mm, minimum height=11mm, align=center}]
\definecolor{acc}{HTML}{4A6FA5}
\node[box] (in) at (0,0) {superposed system\\plus ready pointer};
\node[box] (uni) at (4.9,1.4) {entangled superposition\\of pointer readings};
\node[box, draw=acc, text=acc] (def) at (4.9,-1.4) {one sharp reading};
\draw[->, black, thick] (in) -- (uni) node[midway, above, sloped, black!70] {unitary};
\draw[->, acc, thick] (in) -- (def) node[midway, below, sloped] {collapse};
\end{tikzpicture}
$$

Written in the density operator, environmental entanglement makes the off-diagonal coherences of a
superposition unobservable in practice, explaining why interference between pointer
states is never seen — while noting that decoherence reframes the problem rather
than removing the postulate of a definite single outcome. For all working purposes
the five postulates are the operating manual of the theory, and the next lessons
apply them to the continuous observables of position and momentum.

[^sakurai-meas]: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ 3rd ed., §1.4 — measurements as filtering by eigenstate, the selective and non-selective forms, expectation values, and the compatibility of observables. Cambridge, 2021.
[^ctdl-post]: **Cohen-Tannoudji, Diu & Laloë**, _Quantum Mechanics_ Vol. I, Ch. III — the postulates stated in full, the physical interpretation of the projection, and the spectral decomposition of the probability. Wiley, 1977.