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Advances in physics theories and applications

Solution of Harmonic Oscillator Perturbed by Electric Field

Habtamu Dagnaw

Published Jun 1, 2019• 1 citation
Publisher

Abstract

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

DOI
10.7176/apta/78-01

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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
The concept of harmonic oscillator particularly one dimensional (1-D) is mentioned in literature repeatedly and is explained in more complex manner by using various methods. This creates difficulties in understanding the description of the concept for new learners. The purpose of this article was to enlighten different methods to formulate harmonic oscillator in improving knowledge about detailed steps to derive eigen energy values in more comprehensible manner for the beginners. The energy values are derived by using classical method, quantum mechanically, Schrodinger time independent equation, perturbation technique, variation method, WKB approximation etc. A coherent way of derivation of eigen values using various approaches makes this article as unique.
This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).
This research investigates the numeric solution of the time-independent Schrödinger equation for the quantum harmonic oscillator by finite difference approach. The harmonic oscillator, described by a quadratic function potential, is a fundamental model in quantum mechanics due to its broad applications, ranging from molecular vibrations to quantum field theory. The time-independent Schrödinger equation is a second-order differential equation that typically poses challenges when solved analytically for complex potentials. The finite difference method become an attractive choice as it transforms the continuous differential equation into a system of linear equations that can be computationally solved through computer programming code. In this study, the spatial domain is discretized, and the second derivative is calculated by using central differences, transforming the TISE into a tridiagonal matrix representing Hamiltonian of system. By finding solutions to this matrix eigenvalue problem, wavefunctions and eigenvalues are obtained. The study results demonstrate that the finite difference approach effectively solves the TISE for the harmonic oscillator. The results obtained by using the finite difference method closely approximate the analytical results. The linear regression results show respectively that the gradient (β1), regression coefficient (β0) and coefficient of determination (R²) approach ideal values of 1, 0, and 1. The z-test results also show that the value of calculated z < critical z, indicating that the wavefunction and probability density, whether estimated by using finite difference approach or analytical methods, are equivalent with confidence level of 95 percent.
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.