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Journal of Huaibei Industry Teachers College

The Solution of the Harmonic Oscillator with Electric Charge at the Electric Field in the Coordinate Basis and Energy Basis

Tao Cai-de

Published Jan 1, 2005• 0 citations
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Abstract

The paper discussed the solution of the harmonic oscillator with electric charge at the electric field in the coordinate basis and energy basis,and used the different method to deduced the eigenvalue and eigenfunction.Point that we not only find the approximate solution by using the perturbation method,but also can find the precise solution.

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The paper discussed the solution of the harmonic oscillator with electric charge at the electric field in the coordinate basis and energy basis,and used the different method to deduced the eigenvalue and eigenfunction.Point that we not only find the approximate solution by using the perturbation method,but also can find the precise solution.
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.
The aim of this work is to analyze the energy eigen value of harmonic oscillator perturbed by electric field. In order to carry out the analysis, we have used two independent methods, suchas the perturbation method and time independent schrodinger equation. We have calculated the first order energy correction, second order energy correction , and the energy eigen value for time independent schrodinger equation. It has been found that the energy eigen value of harmonic oscillator is the same in two independent methods. Keywords: Harmonic ocillator, Electric Field, Perturbation Method , Eigen value. DOI : 10.7176/APTA/78-01 Publication date :June 30 th 2019
To understand the time-dependent process in quantum mechanics, an exact expression for the dynamical change of the transition probability and the expectation value of the oscillation energy are presented for the system of the charged harmonic oscillator interacting with an external field through the electric dipole interaction of either type A・p or E・r. The oscillating wave packet is shown to be made from an eigenfunction of the free oscillator Hamiltonian by the interaction during some time. The off-resonance, difference between the field frequency and the proper frequency of the oscillator, controls the number of basis states significant in the transient process. Even on complete resonance, the golden rule holds true only for a short duration and all other nonresonant basis states participate in turn as the interaction time elapses. The difference between two types of dipole interaction exists certainly in the probability distribution in the basis states and the oscillation energy, though it does not in the total current of the oscillator.