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 . We consider the potential
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
. 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.