= Hermite function
{c}
{title2=$H_n(x)=\frac{e^{-x^2/2}}{\pi^{1/4}\sqrt{2^nn!}}(-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}$}
The normalized Hermite functions multiply the physicists' <Hermite polynomials> by a Gaussian. They form an <orthonormal basis> of $L^2(\mathbb R)$ and satisfy $(-d^2/dx^2+x^2)H_n=(2n+1)H_n$. With $a=(x+d/dx)/\sqrt2$ and $a^*=(x-d/dx)/\sqrt2$, they are $(a^*)^nH_0/\sqrt{n!}$ and the <quantum harmonic oscillator> is $2a^*a+1$. For completeness, if $u$ is orthogonal to all of them, all moments of $u(x)e^{-x^2/2}$ vanish. Its transform $\int u(x)e^{-x^2/2}e^{zx}\,dx$ is entire by <Cauchy-Schwarz inequality>, has every derivative zero at zero, and hence vanishes identically. Uniqueness of the <Fourier transform> gives $u=0$.
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