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Journal of Xinyang Normal University

Study on Perturbed Harmonic Oscillator by Particle-Number Representation

Xie Quan

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

Perturbed harmonic oscillator's approximate values of energy level and wave function could be easily calculated by particle-number representation.The proccess is very complicated if it is calculated by coordinate representation.The applicability to solve perturbation of harmonic oscillator by particle-number representation is ascertained due to the study on a example.

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Perturbed harmonic oscillator's approximate values of energy level and wave function could be easily calculated by particle-number representation.The proccess is very complicated if it is calculated by coordinate representation.The applicability to solve perturbation of harmonic oscillator by particle-number representation is ascertained due to the study on a example.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.
The non-degeneracy perturbation theory was investigated in the Torres-Vega and Frederick(T-F) phase space.On the basis of one perturbation,the second and third perturbation formulas of energies and function were deduced.Contrast the probability density of coordinate and momentum with phase space by harmonic oscillator,show that the quantum phase space can get more information than coordinate representation's or momentum representation's when study the perturbation of system in quantum phase space.
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.