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

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

This Demonstration calculates eigenvalues and eigenfunctions for the perturbed Schrödinger equation with , where (x)=. Units are . The energies and wavefunctions for the unperturbed potential (x) are given by =n+ and (x)=(x)n!, where (x) is a Hermite polynomial. When you select "", the numerical solution for and the unperturbed solution (x) are plotted. When you select "", is plotted. When you select "", is shown as a solid black line, (x) as a dashed red curve and as a blue curve. The unperturbed eigenvalue is given by =n+ in all cases.

-ψ+(V(x)-E)ψ=0

2

ℏ

2m

2

2

x

V(x)=(x)+α

(0)

V

β

x

(0)

V

1

2

2

ω

2

x

ℏ=m=ω=1

(0)

V

(0)

E

1

2

(0)

ψ

-2

2

x

H

n

1/4

π

n

2

H

n

ψ&

(0)

ψ

ψ(x)

(0)

ψ

Δψ

ψ(x)-(x)

(0)

ψ

V(x)

V(x)

(0)

V

ψ(x)

E

n

1

2