The energies of a 1D quantum mechanical harmonic oscillator are
$\begin{align*}E_n = \left(n + 1/2 \right)\hbar \omega = \left(n + 1/2 \right)h \nu \end{align*}$
where $n = 0,1,2,3,4,...$ . Therefore, the ground state energy is
$\begin{align*}\frac{1}{2} h \nu\end{align*}$

Solution to 1992 Problem 56