Oscillator length
= Oscillator length
{title2=$\ell=\sqrt{\hbar/(m\omega)}$}
The oscillator length fixes the spatial scale of a <quantum harmonic oscillator> of mass $m$ and angular frequency $\omega$. Its normalized ground <wavefunction> is $\varphi_0(x)=\pi^{-1/4}\ell^{-1/2}\exp[-x^2/(2\ell^2)]$, so $|\varphi_0(0)|^2=1/(\sqrt\pi\ell)$ and $\langle x^2\rangle=\ell^2/2$. The definition of $\ell$ is distinct from the root-mean-square displacement $\ell/\sqrt2$.