In this talk, I will discuss the ground state eigenfunction of a class of Schrödinger operators on a convex planar domain. We will see how to construct two length scales and an orientation of the domain defined in terms of eigenvalues of associated differential operators. These length scales will determine the shape of the intermediate level sets of the eigenfunction, and as an application allow us to deduce properties of the first Dirichlet eigenfunction of the Laplacian for a class of three dimensional convex domains. In the two dimensional case, with constant potential, we will see that the eigenfunction satisfies a quantitative concavity property in a level set around its maximum, consistent with the shape of its intermediate level sets.