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Journal of Anshan Normal University

The Solutions of One dimensional Harmonic Oscillator Energy Eigenvalues in Different Representation

Xiao Liu

Published Jan 1, 2000• 0 citations
OpenAlex

Abstract

The author gives four approaches of calculating the energy eigenvalues of one dimensional harmonic oscillator.They are respectively the method of finding the solution in coorinate representation,the method of finding the solution in momentum representation;the method of finding the solution in energy representation,and the method of finding the solution by direct vectors.\;

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The author gives four approaches of calculating the energy eigenvalues of one dimensional harmonic oscillator.They are respectively the method of finding the solution in coorinate representation,the method of finding the solution in momentum representation;the method of finding the solution in energy representation,and the method of finding the solution by direct vectors.;
The one-dimensional harmonic oscillator is one of the few models in quantum mechanics that can be solved analytically, and it is used to exam the theories and methods of quantum mechanics, possessing rich connotations. Using the Fourier transform and the generating function of Hermite polynomials, the transformation from the eigenfunction in the coordinate representation to that in the momentum representation for a one-dimensional harmonic oscillator has been achieved. The equivalence and connection between the two representations for describing a one-dimensional harmonic oscillator are verified. The eigenenergies and eigenfunctions of the one-dimensional harmonic oscillator in the coordinate representation and momentum representation are also obtained by using the algebraic solution method, respectively. The equivalence of the two representations is further discussed. The idea of psolving physical problems in higher dimensional spaceq is emphasized, which deepens students' understanding of quantum mechanics and stimulates their interest in learning.
By means of quadratic theory,we present a general method to solve the energy eigenvalue of coordinate and momentum coupling of n-dimensional harmonic oscillator.To apply this method to solve the energy eigenvalue of two-dimensional and three-dimensional coupling harmonic oscillator,our conclusion is the same with the others.Since the present method does not require a specific form of transformation matrix,this makes it be simple to solve the eigenvalue of a symmetrical form Hamilton and easy to calculate the results.This method is a universal and very effective algebraic method.
Linear harmonic oscillator takes an important role in quantum mechanics.Many systems can be taken approximately as of linear harmonic oscillator.Consequently,the study on linear harmonic oscillator is significant in many ways.Using methods of formula and perturbation,this paper finds out the energy eigenvalue of one-dimensional linear harmonic oscillator in weak electric field,and presents clearly two typical solutions for the energy level of one-dimensional linear harmonic oscillator in weak electric field.