= Solution
Put $\gamma=m\omega/\hbar$. Substitution of $\psi=F(y)e^{-\gamma y^2/2}$ in the one-dimensional <Schrödinger equation> with energy $(n+1/2)\hbar\omega$ gives
$$
F''-2\gamma yF'+2\gamma nF=0.
$$
For degrees zero, one and two, coefficient comparison yields, up to independent nonzero constants,
$$
\boxed{F_0=1,\qquad F_1=y,\qquad F_2=y^2-\frac1{2\gamma}.}
$$
For the shifted coordinates $y_i=x_i-(r_0)_i$, let $G(y)=e^{-\gamma|y|^2/2}$. A basis of <wave functions> for the first three energy levels is:
* $N=0$: $\boxed{G}$.
* $N=1$: $\boxed{y_1G,\ y_2G,\ y_3G}$.
* $N=2$: $\boxed{(y_i^2-1/(2\gamma))G\ (i=1,2,3),\quad y_1y_2G,\ y_1y_3G,\ y_2y_3G}$.
These are products of the one-dimensional <polynomial> solutions. Arbitrary linear combinations within each listed eigenspace are also stationary states. Their counts $1,3,6$ agree with part (a), so the degeneracies have all been included.
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