---
tag: 52VS07H6GK
kind: concept
name: Hermite functions
msc: [33C45, 42C10, 81Q05]
sources:
  - url: https://en.wikipedia.org/wiki/Hermite_polynomials
---

## Summary

The Hermite functions $\psi_n$ are the [[Hermite polynomials|MGACY98E9H]] multiplied by the Gaussian $e^{-x^2/2}$ and normalised.
They form an [[orthonormal basis|0S9GGTR0EX]] of the [[Hilbert space|HM2HHE3J51]] $L^2(\R)$.
Each $\psi_n$ is an eigenfunction of the [[Fourier transform|DPVM34ADZV]] and of the quantum harmonic oscillator.

## Explanations

### L1 human

A Hermite polynomial grows without bound, so it cannot describe a wave that dies out far away.
Multiplying by the bell curve $e^{-x^2/2}$ makes it decay fast in both directions.
The first Hermite function is a bell curve itself, $\psi_0(x) = \pi^{-1/4} e^{-x^2/2}$.
The next ones are bell curves with more and more wiggles, and in quantum mechanics they are the states of a particle on a spring.