Exact Solution for Rectangular Double-Well Potential

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.