# Harmonic-Gaussian Double-Well Potential

Harmonic-Gaussian Double-Well Potential

A variant of a double-well potential is a harmonic oscillator perturbed by a Gaussian, represented by the potential . A similar function was used to model the inversion of the ammonia molecule [1]. The problem can be treated very efficiently using second-order perturbation theory based on the unperturbed harmonic oscillator. The first six energy levels are computed here.

V(x)=k+α

1

2

2

x

-β

2

x

e

The unperturbed Hamiltonian is =-+k. For convenience, we set . The unperturbed eigenfunctions are given by (x)=(x), where (x) are Hermite polynomials. The unperturbed eigenvalues are then =n+,n=0,1,2,…. The perturbation is the Gaussian function =β. To second-order in perturbation theory, we have =++-. (In practice, the sum is truncated at some ).

(0)

H

2

ℏ

2μ

2

d

d

2

x

1

2

2

x

ℏ=μ=k=1

ϕ

n

-1/2

(n!

n

2

π

)H

n

-2

2

x

H

n

(0)

E

n

1

2

(1)

H

-α

2

x

e

E

n

(0)

E

n

(1)

H

n,n

∞

∑

k≠n=0

(1)

H

n,k

(1)

H

k,n

(0)

E

n

(0)

E

k

n

max

When , the perturbation reduces to zero and the system reverts to a simple harmonic oscillator. As the central barrier becomes wider, the lower eigenvalues ( and ; also and ) approach degenerate pairs. The situation becomes similar to tunneling, giving two eigenstates of opposite parity, with their linear combinations approximating localized states.

α=0

n=0

n=1

n=2

n=3