Perturbation Theory Applied to the Quantum Harmonic Oscillator

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 quartic perturbation, the lowest-order correction to the energy is first order in , so that , where . For a cubic perturbation, the first-order correction vanishes and the lowest-order correction is second order in , so that , where .

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