3.7 The postulates of quantum mechanics

By now the tools are all in place. This section does a job of tidying up: compress the entire theory into the fewest possible assumptions.

It is worth being clear about why this matters. At this point you hold a pile of facts: the Born rule, the Schrödinger equation, operators, eigenvalues, collapse… They do not all have the same standing — some are starting points that must simply be granted, others follow from those starting points. Fail to draw that line, and you end up memorising derivable results as if they were mysterious decrees.

The five postulates

Postulate 1: states

The state of an isolated physical system is completely described by a normalised vector $\ket{\psi}$ in some Hilbert space $\mathcal H$ .

\ket{\psi}\in\mathcal H,\qquad \braket{\psi}{\psi}=1\tag{3.7.1}

Two vectors differing by a global phase, $\ket{\psi}$ and $\ee^{\ii\alpha}\ket{\psi}$ , describe the same physical state.

The words “completely described” carry real content: they assert that no other, finer description exists — no extra variables hiding underneath. This is not self-evident; it is an assumption that experiment can test, and the Bell tests of section 9.4 test exactly this.

Experimental evidence: every manifestation of the superposition principle, from double-slit interference to quantum computing.

Postulate 3: measurement (Born rule + projection)

With the system in $\ket{\psi}$ , the probability that measuring $\hat A$ yields the eigenvalue $a_n$ is

P(a_n)=|\braket{a_n}{\psi}|^2\tag{3.7.3}

(In the degenerate case, $P(a_n)=\|\hat P_n\ket{\psi}\|^2$ , with $\hat P_n$ the projector onto that eigenspace.)

Immediately after the measurement, the state becomes

\ket{\psi}\ \longrightarrow\ \frac{\hat P_n\ket{\psi}}{\|\hat P_n\ket{\psi}\|}\tag{3.7.4}

Experimental evidence: the randomness of individual measurement outcomes, the convergence of frequencies to $|c_n|^2$ over many repetitions, and the fact that an immediately repeated measurement always returns the same result. Section 3.8 is devoted to this postulate.

What is a corollary, not a postulate

This list matters just as much as the postulates themselves.

The picture

All of these are derived:

Eigenvalues are real ← Hermiticity in postulate 2 (section 3.5)
Eigenvectors of distinct eigenvalues are orthogonal ← same source
Eigenvectors form a complete basis ← the spectral theorem
$\sum_n|c_n|^2=1$ ← completeness + normalisation
The uncertainty principle ← postulates 1, 2 + commutation relations (section 3.6)
Expectation values in a stationary state are time-independent ← postulate 4
Probability conservation / the continuity equation ← postulate 4 + Hermiticity of $\hat H$
Ehrenfest’s theorem ← postulates 3, 4

The mathematics

These are genuine assumptions:

States are vectors in a Hilbert space (why complex numbers?)
Observables are Hermitian operators
$|\braket{a_n}{\psi}|^2$ is a probability (why the modulus squared?)
Evolution is generated by $\hat H$
Identical particles come with only two exchange symmetries

The two “why”s in parentheses still have no generally accepted derivation from anything more fundamental. There are beautiful partial results — Gleason’s theorem proves that in dimension $\ge3$ , any probability assignment satisfying basic consistency requirements must take the Born-rule form — but it presupposes the Hilbert-space structure itself.

Do postulates 3 and 4 contradict each other?

This is quantum mechanics’ most famous fault line, and it pays to see it clearly early.

	Postulate 4 (evolution)	Postulate 3 (measurement)
Linear?	yes	no (renormalisation required)
Unitary?	yes	no
Reversible?	yes	no
Deterministic?	yes (the state’s evolution is fully determined)	no (outcomes are random)
Continuous?	yes	no (an instantaneous jump)

The two postulates have exactly opposite properties. And the theory never says: when do you use which?

“Measurement” enters the postulates as an undefined primitive. A measuring apparatus is itself made of atoms and should obey postulate 4. So at which step does the collapse actually happen?

This is the measurement problem. It is not a technical difficulty that finer calculation can dissolve, but a seam in the structure of the theory. The disagreements among interpretations of quantum mechanics concentrate almost entirely here:

Copenhagen: accept the seam, treat measurement as primitive, ask no further.
Many-worlds: delete postulate 3 and keep only postulate 4, at the price of all branches actually existing.
Decoherence: explains why collapse appears to happen (section 10.5), but still needs an extra story for why we see only one outcome.
Bohmian mechanics / GRW: modify the theory itself, each paying its own price.

Section 3.8 and section 13.7 discuss this in more detail. What to remember here: this is not something you failed to understand — this is simply where the subject stands.

A common puzzle: where did the wavefunction go?

A frequent question at this point: how come the five postulates never mention the wavefunction?

Because the wavefunction is derived. By postulate 1 the state is $\ket{\psi}$ ; by postulate 2 position is an observable $\hat x$ with a set of (generalised) eigenstates $\{\ket{x}\}$ ; hence

\psi(x)\equiv\braket{x}{\psi}\tag{3.7.7}

The wavefunction is just the state’s components in the position basis — nothing more. Chapter 2 started from it because it is concrete and can be drawn; but in the logical structure it is not a fundamental object.

Likewise, Schrödinger’s partial differential equation is postulate 4 written out in the position basis:

\ii\hbar\frac{\partial}{\partial t}\braket{x}{\psi}=\bra{x}\hat H\ket{\psi} =-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi\tag{3.7.8}

Key formulas

Postulate 1: states
$\ket{\psi}\in\mathcal H,\quad\braket{\psi}{\psi}=1$
Global phase has no physical meaning; "completely described" rules out hidden variables
Postulate 2: observables
$\hat A^\dagger=\hat A,\quad\text{measured values}\in\{a_n\}$
Realness and orthogonality are corollaries, not extra assumptions
Postulate 3: Born rule
$P(a_n)=|\braket{a_n}{\psi}|^2$
Why the modulus squared — no accepted deeper reason to this day
Postulate 3: projection
$\ket{\psi}\to\hat P_n\ket{\psi}/\|\hat P_n\ket{\psi}\|$
Nonlinear, non-unitary, irreversible
Postulate 4: evolution
$\ii\hbar\frac{\dd}{\dd t}\ket{\psi}=\hat H\ket{\psi}$
Linear, unitary, reversible, deterministic
Postulate 5: identity
$\hat P_{12}\ket{\psi}=\pm\ket{\psi}$
+ for bosons, − for fermions

Self-check3 questions

What comes next

Of the five postulates, the most unruly is the third. The next section takes it apart on its own: what the projection postulate actually says, what it does not say, and why it sparked seventy years of debate.

The postulates of quantum mechanics

Recommended first

After this section you should be able to

The five postulates

What is a corollary, not a postulate

The picture

The mathematics

Key formulas

Self-check3 questions

What comes next

The five postulates

What is a corollary, not a postulate

◑The picture

∑The mathematics

∎Key formulas

What comes next

The picture

The mathematics

Key formulas