Exact Solution for Rectangular Double-Well Potential
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 -⩽x⩽+, 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 (x)=sinnx with =/2, 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.
V(x)=∞
x<0
x>π
V(x)=
V
0
π
2
a
2
π
2
a
2
V(x)=0
ℏ=m=1
-ψ''(x)+V(x)ψ(x)=Eψ(x)
1
2
ψ
n
1/2
(2/π)
E
n
2
n
n=1,2,3,…
n=8
n=1
n=2
ψ
1
ψ
2
ψ
1
ψ
2
|L〉
Be forewarned that plotting a piecewise-continuous eigenfunction might take some time.