Exact Solution for Rectangular Double-Well Potential
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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.
Contributed by: S. M. Blinder (May 2013)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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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