Definition

Normalization condition

\int_{- \infty}^{\infty} ∣ ψ (x) ∣^{2} d x = 1

Statement

The total probability of finding the particle somewhere in space must equal one.

Symbols and units

$ψ (x)$	wave function	—
$∣ ψ ∣^{2}$	probability density	m⁻¹

When it applies

Any valid, physically realizable wave function.

Derivation

Since |ψ|² is a probability density, integrating it over all space must give total probability 1 — the particle must be somewhere:

\int_{- \infty}^{\infty} ∣ ψ (x) ∣^{2} d x = 1

Watch out for

|ψ|² is a probability DENSITY, not a probability itself — it must be integrated over a range to give an actual probability. A value of |ψ(x)|² greater than 1 at some point is entirely possible and not a contradiction.