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Journal of Anyang Normal University

On the Relationships Between Lower Order Energy Levels and Wave Functions in Nondegenerate Perturbation Theory

Niu Xiao-ping

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

This paper is devoted to a systematic discussion of nondegenerate perturbation methods.The energy corrections and state corrections are described respectively.And relationships among them have been studied at the end.

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In this article, for pedagogical purposes we have discussed the application of nondegenerate perturbation theory up to the third order to compute energy eigenvalues and wave functions for the quantum anharmonic oscillator. Energy levels of a single quartic oscillator for 𝜆 values in range of 0.1-1 are given. Perturbed and non-perturbed wave functions of the levels up to the fourth excited level are compared. Ground, first and second excited energy levels are also calculated by applying finite differences method and, results are compared with the ones obtained via perturbation theory. It is found that perturbation theory gives comparable results only for a small 𝜆 parameter and for the ground state. The quartic term in the Hamiltonian of the anharmonic oscillator leads to a more effective confinement of the particle which is deduced from the plots of wavefunctions and probability distributions. Meanwhile, the number of zero crossing nodes of the wavefunctions increases as the energy level increases, which is an expected result for both the harmonic and anharmonic oscillator.