Wave function

Builds on Lp space Quantum state Probability density function Schrödinger equation

Definition

Let $n \ge 1$ , and consider a quantum particle without spin moving in $\R^n$ , with Hilbert space $L^2(\R^n)$ . A wave function of the particle is a measurable function $\psi \colon \R^n \to \C$ such that $\int_{\R^n} \abs{\psi(x)}^2 \, dx = 1.$ It describes the pure quantum state with unit vector $\psi$ in $L^2(\R^n)$ . The function $\abs{\psi}^2$ is the probability density of the position of the particle. Two wave functions equal almost everywhere are the same vector. Two that differ by a constant factor $e^{i\theta}$ , $\theta \in \R$ , give the same state.

Remarks

For $N$ particles in $\R^3$ , a wave function is a unit vector of $L^2(\R^{3N})$ , a function of all the positions together. A time-dependent wave function $\psi(x, t)$ is a family of wave functions that solves the Schrödinger equation.

Example: $\psi(x) = \pi^{-1/4} e^{-x^2/2}$ on $\R$ . Indeed $\int_\R \pi^{-1/2} e^{-x^2}\, dx = 1$ , so $\psi$ is a wave function, and the position is normally distributed with mean $0$ and variance $1/2$ .

Non-example: the plane wave $\psi(x) = e^{ikx}$ with $k \in \R$ has $\abs{\psi}^2 = 1$ , so $\int_\R \abs{\psi}^2 \, dx = \infty$ . It is not in $L^2(\R)$ and is not a wave function, although it is used as a generalised eigenfunction of the momentum.

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Tag	`TSEG9M3QQM`
URI	`m2://TSEG9M3QQM`
Kind	definition
Writing	W0 stub
MSC	81Q05, 81P16
Defines	Wave function
Uses	Lp space, Quantum state, Probability density function, Schrödinger equation
Changed	3 November 2026

Wave function

Definition

Remarks

Reading path

Further reading

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