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We study the first Dirichlet eigenfunction of a class of Schr\"odingeroperators with a convex potential V on a domain $\Omega$. We find two lengthscales $L_1$ and $L_2$, and an orientation of the domain $\Omega$, whichdetermine the shape of the level sets of the eigenfunction. As an intermediatestep, we also establish bounds on the first eigenvalue in terms of the firsteigenvalue of an associated ordinary differential operator.

The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schr\"odinger Operators