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  "body": "\n## Summary\n\nThe Hermite functions $\\psi_n$ are the [[Hermite polynomials|MGACY98E9H]] multiplied by the Gaussian $e^{-x^2/2}$ and normalised.\nThey form an [[orthonormal basis|0S9GGTR0EX]] of the [[Hilbert space|HM2HHE3J51]] $L^2(\\R)$.\nEach $\\psi_n$ is an eigenfunction of the [[Fourier transform|DPVM34ADZV]] and of the quantum harmonic oscillator.\n\n## Explanations\n\n### L1 human\n\nA Hermite polynomial grows without bound, so it cannot describe a wave that dies out far away.\nMultiplying by the bell curve $e^{-x^2/2}$ makes it decay fast in both directions.\nThe first Hermite function is a bell curve itself, $\\psi_0(x) = \\pi^{-1/4} e^{-x^2/2}$.\nThe next ones are bell curves with more and more wiggles, and in quantum mechanics they are the states of a particle on a spring.\n",
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