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One dimensional (1-D) harmonic oscillators revisited

Nilesh P. Barde, Pranav P. Bardapurkar

Published May 1, 2018• 0 citations
OpenAlex

Abstract

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.

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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.
Quantum mechanics is one of the branches of physics. In this mechanics, physical problems are solved by algebraic and analytic methods. By applying a simple procedure we can find the general solutions of Schrodinger’s time - independent wave equation of one dimensional quantum mechanical harmonic oscillator without making any approximation. In this paper, we will discuss the Eigenenergy values and Eigenfunctions of one of the most important physical models of quantum mechanics, namely the one-dimensional Quantum mechanical Harmonic Oscillator by modifying the Hermite differential equation.
The Schrödinger equation is a significant achievement on the development of quantum mechanics. By solving the Schrödinger equation, the fundamental behaviors and properties of a microscopic particle can be found in one- to three-dimensions. The article focuses on the derivations of quantum harmonic oscillators in one dimension by solving second order-differential equation. The wave functions, probability densities under different energy levels are presented. The results can be used to estimate different forms of continuous potential experienced by an oscillator. By calculating the uncertainty relation of the oscillator under one specific excited state, the general relation can be confirmed. Based on the output, by splitting the variables, the two-dimensional harmonic oscillator can also be derived. The degenerate energy levels are presented as . Numerical simulations are made to visualize those results and suggests the relation of energy levels and number of maxima of the probability densities.
Chapter 9 is devoted to a detailed study of the harmonic oscillator in one dimension. The time-dependent Schrödinger equation is used to construct the coherent oscillator packet with minimal dispersion, which provides an opportunity to appreciate the similarities and differences between the dynamics of the classical and quantum oscillators. In a second step, we study the stationary solutions of the oscillator, which represent one of the most important orthogonal sets of quantum eigenfunctions because of its many applications, properties, and relative simplicity. Further, the use of raising and lowering (or creation and annihilation) operators is shown to be a powerful way of dealing with both material and field oscillators in general. Finally, the problem of two coupled oscillators is used to generalize the Schrödinger equation to the case of more than one particle.
Abstract We consider the time-independent Schrödinger equation for the harmonic oscillator potential and determine its bound states and energy levels in one spatial dimension using two approaches: the algebraic and the analytic method. Using the former, we introduce ladder operators: the annihilation and creation operators, as well as the occupation-number operator, and determine the ground state of the harmonic oscillator to be a Gaussian wave packet, while the latter method yields the general bound state solutions in terms of the Hermite polynomials. We further discuss the zero-point energy and uncertainty relation for the quantum harmonic oscillator and make a comparison to the classical harmonic oscillator. Finally, we examine the three-dimensional harmonic oscillator, which leads us to the description of systems with multiple degrees of freedom via the tensor product
By the Fourier transform ation,we change the Schr dinger equation of one-dimensional linear harmonic oscillator to the first-order differential equation which can be easily solved.The progressional solution of the Schr dinger equation can be obtained by the inverse Fourier transform ation after we solve the first-order differential equation.Finally,we can fix the eigen-value and eigen-function of energy based on the boundary condition that wave function is zero when the variable becomes infinite.
The concept of time dependent Schrödinger equation (TDSE) illustrated in literature and even during class room teaching is mostly either complex or meant for advanced learners. This article is intended to enlighten the concept to the beginners in the field and further to improve knowledge about detailed steps for abstract mathematical formulation used which helps in understanding to derive TDSE using various tools and in more comprehensible manner. It is shown that TDSE may be derived using wave mechanics, time independent equation, classical & Hamilton-Jacobi’s equations. Similar attempts have been done earlier by some researchers. However, this article provides a comprehensive, lucid and well derived derivation, derived using various approaches, which would make this article unique.