  We study the first Dirichlet eigenfunction of a class of Schr\"odinger
operators with a convex potential V on a domain $\Omega$. We find two length
scales $L_1$ and $L_2$, and an orientation of the domain $\Omega$, which
determine the shape of the level sets of the eigenfunction. As an intermediate
step, we also establish bounds on the first eigenvalue in terms of the first
eigenvalue of an associated ordinary differential operator.
