Quantum Mechanics Core Recommended

The Postulates of Quantum Mechanics

The four rules everything else in this curriculum is built on

Builds on:Linear Algebra Essentials Complex Numbers & Dirac Notation

By the end of this topic you'll be able to

State the four postulates of quantum mechanics in your own words

Explain why quantum evolution (outside measurement) must be linear and reversible

Distinguish what the postulates say from the many popular (and often misleading) analogies for them

Physics is the quantitative description of natural phenomena, and any such description begins by fixing what can be measured. Associated with every measurement is a spectrum — the set of values it could return — and quantum mechanics attaches to each value in that spectrum a quantum amplitude, a complex number whose modulus-squared gives the probability of obtaining it.

Consequently, knowledge of a system's dynamical state is not a single number, as in classical physics, but an entire set of these amplitudes. Dirac's ket |ψ⟩ is simply a container for a complete set of them, in the same way an ordinary vector is a container for a set of coordinates — useful precisely because it doesn't commit you to any one choice of basis.

States. The state of an isolated system is a unit vector |ψ⟩ in a complex vector space called Hilbert space. Normalization, ⟨ψ|ψ⟩ = 1, is not a bookkeeping convenience — it is exactly the condition that makes the probabilities extracted from |ψ⟩ sum to one.

Polar angle θ90°

Phase angle φ0°

∣ ψ ⟩ = 0.707 ∣0 ⟩ + 0.707 e^{i 0 π} ∣1 ⟩

P(0) = 50%P(1) = 50%

Evolution. Between measurements, a state evolves by a linear, reversible transformation, |ψ(t)⟩ = U(t)|ψ(0)⟩, with U unitary (U†U = I).

Unitarity preserves length, so total probability is conserved, and the transformation can always be undone by applying U†. This is why a quantum gate is reversible in a way that a classical AND or OR gate simply isn't.

Measurement. Measuring an observable yields one of its eigenvalues, with probability given by the modulus-squared amplitude on the corresponding eigenvector, and the state abruptly changes to match — a process known as the collapse of the wavefunction.

Prepare |ψ⟩ = cos(θ/2)|0⟩ + sin(θ/2)|1⟩θ = 60°

Before measuring, $P (0)$ and $P (1)$ above are just probabilities — press Measure to collapse the state to one definite outcome.

It is tempting to read collapse as nothing more than an updating of our knowledge — as if the system had already been in that state, and the measurement only revealed the fact. Binney and Skinner's textbook is blunt about why this reading fails: wavefunction collapse is associated with a real physical disturbance of the system, not merely a change in what we happen to know about it.

Composite systems. Combining two systems A and B into one system AB, the state space of AB is the tensor product of the individual spaces. This is the postulate that makes entanglement possible: a general state of AB need not be expressible as a state of A times a state of B.

Between measurements, none of this is fuzzy. Given |ψ(0)⟩ and the system's Hamiltonian, U(t) is computed exactly and deterministically — quantum mechanics is not a theory of pervasive vagueness. Randomness enters at exactly one place, measurement, and in one precisely quantifiable way.

Try It Yourself

Worked Example

A system is prepared in |ψ⟩ = (1/√3)|e₁⟩ + (√2/√3)|e₂⟩, where |e₁⟩ and |e₂⟩ are the eigenstates of some observable H with eigenvalues λ₁ = 1 and λ₂ = 4. First confirm |ψ⟩ is properly normalized, then find the probability of each measurement outcome and the expectation value ⟨H⟩.

1Check normalization: |1/√3|² + |√2/√3|² = 1/3 + 2/3 = 1. ✓ — the state postulate is satisfied.
2By the Born rule, P(λ₁) = |⟨e₁|ψ⟩|² = |1/√3|² = 1/3, and P(λ₂) = |⟨e₂|ψ⟩|² = |√2/√3|² = 2/3.
3These sum to 1/3 + 2/3 = 1, as they must — every measurement returns exactly one of the two outcomes.
4The expectation value is the probability-weighted average of the possible readouts: ⟨H⟩ = P(λ₁)·λ₁ + P(λ₂)·λ₂ = (1/3)(1) + (2/3)(4) = 1/3 + 8/3 = 3.

Answer

P(λ₁=1) = 1/3, P(λ₂=4) = 2/3, and ⟨H⟩ = 3 — the average result you'd get from measuring many identically-prepared copies of |ψ⟩, even though any single measurement returns only 1 or 4, never 3.

Reference

State	$∣ ψ ⟩, ⟨ ψ ∣ ψ ⟩ = 1$	A unit vector in Hilbert space
Unitary evolution	$U^{†} U = I$	Linear, reversible, probability-preserving evolution
Observable	$H = H^{†}$	A measurable quantity (Hermitian operator); eigenvalues are the possible readouts
Born rule	$P (i) = ∣ ⟨ e_{i} ∣ ψ ⟩ ∣^{2}$	Probability of measuring eigenvalue i
Composite system	$∣ ψ ⟩_{A B} \in H_{A} \otimes H_{B}$	Joint state space is a tensor product

Quick Check

What property must U have for |ψ(t)⟩ = U|ψ(0)⟩ to be valid quantum evolution?

Which postulate is responsible for the possibility of entanglement?

Further Resources

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The Qubit & Superposition