It is possible to derive exact solutions of the Schrödinger equation for an infinite square well containing a finite rectangular barrier, thus creating a double-well potential. The problem was previously approached using perturbation theory [1]. We consider the potential

for

and

,

for

, and

elsewhere. We set

for convenience. Solutions of the Schrödinger equation

have the form of particle-in-a-box eigenfunctions in three connected segments. For the unperturbed problem, the normalized eigenstates are

with

, for

. The computations are spelled out in the Details section. You can display eigenvalues and eigenfunctions up to

. For larger barriers, it will be discovered that

becomes the lowest-energy state, crossing

. We will continue, however, to maintain the quantum numbers of the unperturbed problem.