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Journal of Changchun Institute of Technology

The spectrum calculating of simple harmonic oscillator under perturbation by similar expanding

Zhiming Liu

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

This paper discusses a method of expanding general perturbation at the most probabe location of simple harmonic stationary state and adding the expansion to standard potential. The modified potential is used to cal- culate the spectrum of Harmonic Oscillator under pertur- bation.

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The harmonic oscillator

Efstratios Manousakis

Abstract This chapter discusses the one-dimensional harmonic oscillator. It begins by discussing under what conditions the harmonic approximation can be used. It then transforms the Hamiltonian of the harmonic oscillator in terms of rising and lowering operators and then it finds its spectrum of eigenvalues and eigenstates.

•Nov 1, 2015•0 citations
Trial wave function was used to calculate ground state energy to obtain a relatively accurate value.After adding potential δ to one-dimensional harmonic oscillator,the odd parity stationary states of the harmonic oscillator still held and the even parity stationary states were no longer correct. Therefore,the perturbation theory and variational method were used respectively to calculate the approximate even parity stationary states,and variational method was used to solve Schrodinger equation.Results show that the calculated approximate even parity stationary states match well with the desired value,especially in excited state,the relative error is less than×10-3,and all the approximate values of energy for all levels are higher than desired value,which prove that it is simple and feasible by using perturbation theory or variational method to calculate approximate even parity stationary states with δ potential added.
Abstract We continue to examine the connection between perturbation theory and the Van Vleck unitary transformation. Here we illustrate the formalism derived earlier by applying it to compute the stationary states of the perturbed harmonic oscillator. We find that each solution of the traditional Brillouin‐Wigner perturbation theory equations gives rise to a different unitary transformation which, when operating on the unperturbed ground state, produces one or the other of the perturbed eigenstates. With any of the perturbed states able to be reached by a unitary transformation on the unperturbed ground state, we advise caution in using approximate solutions of the perturbation equations in general cases, lest an unexpected stationary state be obtained.