Wave function

Level AdvancedDifficulty ★★★★★Application⌖ Open in the map

What is it?

A complex-valued function $\psi(x)$ whose squared modulus is a probability density (Born rule): the probability of finding the particle in $[a,b]$ is $\int_a^b|\psi|^2\,\dd x$ , and normalization requires $\int|\psi|^2 = 1$ .

Formulas

P(a \le X \le b) = \int_a^b |\psi(x)|^2\,\dd x, \qquad \int_{-\infty}^{\infty}|\psi|^2\,\dd x = 1

The mathematics behind it

Complex numbers★★★★★fundamental
A wave function takes complex values; only $|\psi|^2$ is a probability.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

ψ Quantum computing and physics

Uncertainty principle★★★★★
Quantum computing★★★★★
Schrödinger equation★★★★★

What depends on it

Uncertainty principle
Quantum computing
Schrödinger equation

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.