We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex, non-negative, potential
V
V
on a convex, planar domain
Ω
\Omega
. In the case where the diameter of
Ω
\Omega
is large and the potential
V
V
varies on different length scales in orthogonal directions, we find two length scales
L
1
L_1
and
L
2
L_2
and an orientation of the domain
Ω
\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.