Harmonic oscillator eigenfunction (${\psi }_n$)

With oscillator length $\ell =\sqrt{\hslash /(m\omega )}$, the eigenfunctions are proportional to $H_n(x/\ell )e^{-x^2/(2{\ell }^2)}$. They have $n$ real nodes, parity $(-1{)}^n$, and energy $\hslash \omega (n+1/2)$.