 Half-harmonic oscillator

A half-harmonic oscillator has $V=m{\omega }^2{x}^2/2$ on $x>0$ and an infinite wall at $x=0$. Its wavefunction vanishes at the wall. Only the odd full-oscillator energy eigenstates satisfy this condition; restricting them to the positive half-line and multiplying by $\sqrt{2}$ restores normalization. Thus $E_r=\hslash \omega (2r+3/2)$ for $r=0,1,\dots$. With $\ell =\sqrt{\hslash /(m\omega )}$, its normalized ground state isIt has no interior node on its allowed half-line, and energy $3\hslash \omega /2$.