The Formal Postulates of Quantum Mechanics

By now you have solved the Schrödinger equation in a box, in a spring, and around a proton; you have measured observables and written states in bra–ket notation. What you have not yet seen is the small, sharp list of assumptions all of that quietly rests on. Every result in quantum mechanics — every energy level, every transition rate, every interference fringe — unspools from just four or five sentences taken as axioms. This page states them plainly and shows how they lock together into the whole subject.

This is the moment quantum mechanics stops being a bag of tricks and becomes a mathematical theory: a state is a vector, an observable is an operator, a measurement is a projection, and time is a rotation. Learn the postulates as a set and the rest of the course reads like theorems that follow from them. Newton needed three laws; quantum mechanics needs about four.

Postulate 1 — states are rays in a Hilbert space

The complete description of a physical system at one instant is a state vector |\psi\rangle, a unit-length element of a complex Hilbert space \mathcal{H}. "Contains everything knowable" is meant literally: if two experimenters agree on |\psi\rangle, they can compute the probability of every possible measurement identically. There is nothing more to know.

Two features of this space do all the work. First it is linear: if |\psi_1\rangle and |\psi_2\rangle are allowed states, so is any combination \alpha|\psi_1\rangle + \beta|\psi_2\rangle — this is the superposition principle, and every quantum weirdness ultimately traces back to it. Second it carries an inner product \langle\phi|\psi\rangle, a complex number measuring how much two states overlap; its squared modulus will be a probability.

One subtlety worth stating precisely: the physical state is really the ray, not the vector. Multiplying by a global phase, |\psi\rangle \to e^{i\gamma}|\psi\rangle, changes no probability and so describes the same physics. A global phase is invisible; relative phases between the pieces of a superposition are everything.

Postulate 2 — observables are Hermitian operators

Every measurable quantity — position, momentum, energy, a spin component — is represented by a Hermitian (self-adjoint) operator \hat{A} = \hat{A}^\dagger acting on \mathcal{H}. The reason it must be Hermitian is not arbitrary mathematical hygiene; it is forced by two physical demands, and both drop out of one theorem.

Its eigenvalues are real: \hat{A}|a\rangle = a|a\rangle with a \in \mathbb{R}. Since these eigenvalues are the only numbers a measurement can return, reality is non-negotiable — a meter cannot read 3+2i grams.
Its eigenvectors form a complete orthonormal basis of \mathcal{H}. So any state can be written as a superposition of the outcomes of measuring \hat{A} — which is exactly what the next postulate needs.

Because the eigenstates \{|a_n\rangle\} are complete, any state expands as

|\psi\rangle = \sum_n c_n\,|a_n\rangle, \qquad c_n = \langle a_n | \psi\rangle,

Postulate 3 — the Born rule and collapse

Here is where randomness finally enters — and it enters only here. Measure the observable \hat{A} on the state |\psi\rangle = \sum_n c_n|a_n\rangle. Then:

The only possible results are the eigenvalues a_n of \hat{A}.
The result a_n occurs with probability P(a_n) = |c_n|^2 = |\langle a_n|\psi\rangle|^2 — the Born rule.
Immediately after a result a_n, the state collapses to the corresponding eigenstate: |\psi\rangle \to |a_n\rangle. Measure again at once and you are certain to get a_n a second time.

The picture below makes this geometric. Take the simplest system, a qubit, with two outcomes |0\rangle and |1\rangle. A state |\psi\rangle = \cos\theta\,|0\rangle + \sin\theta\,|1\rangle is a unit vector; the amplitudes c_0, c_1 are its projections onto the two axes, and the probabilities \cos^2\theta,\ \sin^2\theta are those projections squared. Slide \theta and watch probability slosh from one outcome to the other, always summing to 1 — that is the Pythagorean identity in disguise.

This is the whole of quantum indeterminism in one image. The state is perfectly definite; what is indefinite is the answer to a question you haven't asked yet. The angle \theta is fixed and knowable; which outcome the apparatus records is not.

Postulate 4 — expectation values

With probabilities in hand, the average result of many measurements on identically-prepared systems is an ordinary weighted mean — but it collapses to a strikingly compact operator sandwich:

\langle \hat{A}\rangle = \sum_n a_n\,P(a_n) = \sum_n a_n\,|c_n|^2 = \langle\psi|\hat{A}|\psi\rangle.

That last equality is worth checking once by hand (Example 2 below) — it is the reason the bra–ket sandwich \langle\psi|\hat{A}|\psi\rangle appears everywhere. The spread of results about that average is the variance (\Delta A)^2 = \langle \hat{A}^2\rangle - \langle\hat{A}\rangle^2, the very quantity the uncertainty principle bounds.

Postulate 5 — unitary time evolution

Between measurements, a closed system evolves deterministically and smoothly by the Schrödinger equation

i\hbar\,\frac{d}{dt}|\psi(t)\rangle = \hat{H}\,|\psi(t)\rangle, \qquad |\psi(t)\rangle = \hat{U}(t)\,|\psi(0)\rangle, \qquad \hat{U}(t) = e^{-i\hat{H}t/\hbar}.

Because \hat{H} is Hermitian, the evolution operator \hat{U} is unitary: \hat{U}^\dagger\hat{U} = \hat{I}. Unitarity is exactly the statement that total probability is conserved — \langle\psi(t)|\psi(t)\rangle = 1 for all time, so the particle never leaks away. Quantum dynamics has two utterly different modes: the smooth unitary flow of Postulate 5, and the abrupt random collapse of Postulate 3. The friction between those two is the whole of the "measurement problem."

A genuine puzzle. Postulate 5 is time-reversible and information-preserving: run \hat{U}(t) backwards with \hat{U}(-t) and you recover the past exactly. Postulate 3 is neither — collapse throws information away and cannot be undone. So which is fundamental? The many-worlds view keeps only Postulate 5 and argues collapse is an illusion of entanglement with the apparatus; objective-collapse theories make Postulate 3 a real physical process; the Copenhagen tradition simply draws a pragmatic line between quantum system and classical measuring device. All make the same laboratory predictions — the postulates above — and differ only in what they say is "really" happening. That is why this is philosophy of physics, not (yet) experiment.

Worked examples

Example 1 — read off probabilities. A qubit is prepared in |\psi\rangle = \tfrac{3}{5}|0\rangle + \tfrac{4}{5}|1\rangle. First check normalisation: (3/5)^2 + (4/5)^2 = 9/25 + 16/25 = 1 ✓. Measuring in the \{|0\rangle,|1\rangle\} basis returns 0 with probability |3/5|^2 = 9/25 = 0.36 and 1 with probability 16/25 = 0.64. They sum to 1, as any honest probability distribution must.

Example 2 — the sandwich formula. Let observable \hat{A} have eigenvalues a_0, a_1 on |0\rangle, |1\rangle. Then \hat{A}|\psi\rangle = c_0 a_0|0\rangle + c_1 a_1|1\rangle, and using orthonormality \langle 0|1\rangle = 0,

\langle\psi|\hat{A}|\psi\rangle = |c_0|^2 a_0 + |c_1|^2 a_1 = \sum_n a_n P(a_n) = \langle\hat{A}\rangle.

The abstract sandwich and the down-to-earth weighted average are literally the same number.

Example 3 — repeated measurement (collapse). Measure the qubit of Example 1 and suppose it yields 0. By collapse the state is now exactly |0\rangle. Measure the same observable immediately again: the probability of 0 is now |\langle 0|0\rangle|^2 = 1 — a certainty. The first measurement reset the system; the randomness does not repeat.

The classic trap. Students often think the states |\psi\rangle and e^{i\gamma}|\psi\rangle are physically different because "the numbers changed." They are not. A global phase multiplies the whole vector and cancels out of every probability |\langle a_n|\psi\rangle|^2 — it is unobservable, which is why Postulate 1 speaks of rays.

But do not over-learn the lesson and dismiss all phases. The relative phase inside a superposition is fully physical: \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle) and \tfrac{1}{\sqrt2}(|0\rangle - |1\rangle) are orthogonal — perfectly distinguishable states, differing only by a sign on one term. Measure them along a tilted axis and they give opposite results. Global phase: invisible. Relative phase: the engine of interference.