  We consider Gaussian random eigenfunctions (Hermite functions) of fixed
energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf
R}^n$. We calculate the expected density of zeros of a random eigenfunction in
the semi-classical limit $h \to 0.$ In the allowed region the density is of
order $h^{-1},$ while in the forbidden region the density is of order
$h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this
difference in "frequency" between the allowed and forbidden nodal sets.
