Eigenvalues and Eigenfunctions for the Harmonic Oscillator with Quartic, Sextic and Octic Perturbations

perturbation exponent β

4

6

8

perturbation prefactor α

0.1

quantum number n

0

ψ

n

&

(0)

ψ

n

Δψ

V(x)

ψ

n

(0)

ψ

n

This Demonstration calculates eigenvalues and eigenfunctions for the perturbed Schrödinger equation

-

2

ℏ

2m

2



ψ



2

x

+(V(x)-E)ψ=0

with

V(x)=

(0)

V

(x)+α

β

x

, where

(0)

V

(x)=

1

2

ω

2

x

. Units are

ℏ=m=ω=1

. The energies and wavefunctions for the unperturbed potential

(0)

V

(x)

are given by

(0)

E

=n+

1

2

and

(0)

ψ

(x)=

-

2

x

2



H

n

(x)

1/4

π

n

2

n!

, where

H

n

(x)

is a Hermite polynomial. When you select "

ψ&

(0)

ψ

", the numerical solution for

ψ(x)

and the unperturbed solution

(0)

ψ

(x)

are plotted. When you select "

Δψ

",

ψ(x)-

(0)

ψ

(x)

is plotted. When you select "

V(x)

",

V(x)

is shown as a solid black line,

(0)

V

(x)

as a dashed red curve and

ψ(x)

as a blue curve. The unperturbed eigenvalue is given by

E

n

=n+

1

2

in all cases.