= Half-harmonic oscillator
{title2=$E_r=\hbar\omega(2r+3/2)$}
A <half-harmonic oscillator> has $V=m\omega^2x^2/2$ 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 $\sqrt2$ restores normalization. Thus $E_r=\hbar\omega(2r+3/2)$ for $r=0,1,\ldots$. With $\ell=\sqrt{\hbar/(m\omega)}$, its normalized <ground state> is
$$
\psi_h(x)=\frac{2x}{\pi^{1/4}\ell^{3/2}}e^{-x^2/(2\ell^2)}\quad(x>0),\qquad \psi_h(x)=0\quad(x<0).
$$
It has no interior node on its allowed half-line, and energy $3\hbar\omega/2$.
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