For a particle moving in continuous space, its quantum state can be represented by a wavefunction

$$\psi(x,t),$$

a complex-valued amplitude associated with position $x$ at time $t$.

Position probability

The Born rule states that

$$|\psi(x,t)|^2$$

is a probability density for position. The probability of finding the particle between $a$ and $b$ is

$$P(a\le x\le b)=\int_a^b|\psi(x,t)|^2,dx.$$

A normalized wavefunction satisfies

$$\int_{-\infty}^{\infty}|\psi(x,t)|^2,dx=1.$$

Phase matters before measurement

Two wavefunctions can have the same magnitude $|\psi|$ but different phases. Their immediate position probabilities are the same, yet their later interference or dynamical behavior can differ.

Wavefunction is not a material wave density

$\psi$ itself is generally complex and is not directly an ordinary measurable density. Observable position probabilities come from its squared magnitude.

Localized and spread-out states

A narrow $|\psi|^2$ describes a state with relatively localized position, while a broad distribution represents greater position uncertainty.

The wavefunction turns the abstract idea of a quantum state into a spatial amplitude whose evolution can be described by the Schrödinger equation.