# 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.