Exact Solution for Rectangular Double-Well Potential

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 for the barrier problem are spelled out in the Details section. You can display eigenvalues and eigenfunctions up to . As the barrier increases in height and width, the and levels approach degeneracy. The linear combinations and then approximate the localized states and |R〉, respectively.
Be forewarned that plotting a piecewise-continuous eigenfunction might take some time.


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Snapshot 1: unperturbed particle-in-a-box eigenstates
Snapshots 2, 3: for larger barriers, the and levels approach degeneracy, as do, to a lesser extent, the and levels
Even solutions, with , have the form
for (which fulfills the boundary condition ),
for (which is even about ),
for (which fulfills the boundary condition ).
Odd solutions, with , have the form
for (same as the even solutions),
for (which is odd about ),
for (which also fulfills the boundary condition ).
In regions I and III, the energy eigenvalues follow from . In region II, we find . Since these energies must be equal, .
The connection formulas for the two region boundaries are most conveniently expressed in terms of the logarithmic derivatives. At , for example, , and analogously for and . This leads to the transcendental equations:
for the even eigenstates, and
for the odd eigenstates, .
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