Postulate 1: State Vectors

The postulates of quantum mechanics are a short list of rules that tell us how to describe a physical system, what we can measure, and how the description changes in time. They are not derived from anything more basic; they are the axioms we adopt because they reproduce experiment. This module works through all five. We begin with the most fundamental: what is the state of a quantum system?

The state-space postulate

Postulate 1. Associated to any isolated physical system is a complex inner-product space (a Hilbert space) $\mathcal{H}$ called the state space. The system is completely described by its state vector, a unit vector $|\psi\rangle \in \mathcal{H}$ .

Three words in this statement carry the entire content, so let us unpack each.

Complex vector space

States live in a vector space over the complex numbers. This means states can be added and scaled by complex numbers, and the results are again valid (unnormalized) states. If $|\psi\rangle$ and $|\phi\rangle$ are states, then so is any superposition

\alpha|\psi\rangle + \beta|\phi\rangle, \qquad \alpha, \beta \in \mathbb{C}.

This linearity is the source of every distinctly quantum phenomenon — interference, entanglement, and the exponential size of the state space all follow from it.

Inner product

The space carries an inner product $\langle\phi|\psi\rangle$ , a complex number that is linear in the ket $|\psi\rangle$ , conjugate-linear in the bra $\langle\phi|$ , and satisfies $\langle\psi|\phi\rangle = \langle\phi|\psi\rangle^{*}$ with $\langle\psi|\psi\rangle \geq 0$ . The inner product gives us lengths and angles: it is what lets us talk about probabilities and orthogonality. The norm of a state is

\big\||\psi\rangle\big\| = \sqrt{\langle\psi|\psi\rangle}.

Unit vector

The physical state must be normalized:

\langle\psi|\psi\rangle = 1.

This is the condition that probabilities sum to one, as we will see when we reach the Born rule. A state that is not normalized describes the same physics once divided by its norm.

A qubit as the simplest example

The smallest nontrivial state space is two-dimensional, $\mathcal{H} = \mathbb{C}^2$ . Its standard orthonormal basis is written $\{|0\rangle, |1\rangle\}$ , and a general qubit state is

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1,

with $\alpha, \beta \in \mathbb{C}$ . The normalization condition $|\alpha|^2 + |\beta|^2 = 1$ is exactly $\langle\psi|\psi\rangle = 1$ when $\{|0\rangle, |1\rangle\}$ are orthonormal.

For an $n$ -qubit system the state space is $\mathbb{C}^{2^n}$ , with $2^n$ basis states $|0\ldots0\rangle, \ldots, |1\ldots1\rangle$ . The dimension grows exponentially in the number of qubits — the reason simulating quantum systems on classical computers is hard.

Global phase carries no physics

Two state vectors that differ only by an overall complex phase,

|\psi\rangle \quad\text{and}\quad e^{i\gamma}|\psi\rangle,

describe the same physical state. No measurement can distinguish them, because (as we will prove with the Born rule) every observable prediction depends only on quantities like $|\langle\phi|\psi\rangle|^2$ , in which the global phase $e^{i\gamma}$ cancels. The genuinely physical object is therefore the ray — the equivalence class of unit vectors related by a global phase — but in practice we work with a representative unit vector and simply remember that the overall phase is unobservable.

Why a vector, and why complex?

Representing states as vectors makes superposition automatic: linear combinations are built into the structure. Making the field complex is forced by dynamics — the Schrödinger equation has an explicit factor of $i$ , so real amplitudes cannot remain real under time evolution. The inner product then supplies the bridge from these complex amplitudes to real, non-negative probabilities. Postulate 1 fixes the arena; the remaining postulates tell us what happens inside it.