WOLFRAM|DEMONSTRATIONS PROJECT

Exact Solution for Rectangular Double-Well Potential

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barrier width a
0.75
barrier height
V
0
6
show wavefunction
n
1
2
3
4
5
6
7
8
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
V(x)=∞
for
x<0
and
x>π
,
V(x)=
V
0
for
π
2
-
a
2
⩽x⩽
π
2
+
a
2
, and
V(x)=0
elsewhere. We set
ℏ=m=1
for convenience. Solutions of the Schrödinger equation
-
1
2
ψ''(x)+V(x)ψ(x)=Eψ(x)
have the form of particle-in-a-box eigenfunctions in three connected segments. For the unperturbed problem, the normalized eigenstates are
ψ
n
(x)=
1/2
(2/π)
sinnx
with
E
n
=
2
n
/2
, for
n=1,2,3,…
. The computations for the barrier problem are spelled out in the Details section. You can display eigenvalues and eigenfunctions up to
n=8
. As the barrier increases in height and width, the
n=1
and
n=2
levels approach degeneracy. The linear combinations
ψ
1
+
ψ
2
and
ψ
1
-
ψ
2
then approximate the localized states
|L〉
and |R〉, respectively.
Be forewarned that plotting a piecewise-continuous eigenfunction might take some time.