 Oscillator length

The oscillator length fixes the spatial scale of a quantum harmonic oscillator of mass $m$ and angular frequency $\omega$. Its normalized ground wavefunction is ${\phi }_0(x)={\pi }^{-1/4}{\ell }^{-1/2}\exp [-x^2/(2{\ell }^2)]$, so $|{\phi }_0(0){|}^2=1/(\sqrt{\pi }\ell )$ and $⟨x^2⟩={\ell }^2/2$. The definition of $\ell$ is distinct from the root-mean-square displacement $\ell /\sqrt{2}$.