The unperturbed energy levels and eigenfunctions of the quantum harmonic oscillator problem, with potential energy

, are given by

and

, where

is the

Hermite polynomial. This Demonstration studies how the ground-state (

) energy shifts as cubic

and quartic

perturbations are added to the potential, where

characterizes the strength of the perturbation. The left graphic shows unperturbed

(blue dashed curve) and the perturbed potential

(red), and the right graphic shows

(blue dashed curve) along with an approximation to the perturbed energy

(red) obtained via perturbation theory. For a cubic perturbation, the lowest-order correction to the energy is first order in

, so that

, where

. For a quartic perturbation, the first-order correction vanishes and the lowest-order correction is second order in

, so that

, where

.