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CRC Press eBooks

One-particle Quantum Harmonic Oscillator

Caio Lima Firme

Published Mar 21, 2022• 0 citations
Publisher

Abstract

This chapter begins discussing the classical harmonic oscillator and its solution as a second-order differential equation of the general type. Next, the one-particle quantum harmonic oscillator is presented, its corresponding Hamiltonian functionand wave equation followed by the undimensionalization of this wave equation yielding a second-order differential equation of the general type which leads to the general Hermite equation and Hermite polinomials as partial solution of the wave function. The energy for each wave function is obtained from the recursion relation of the Hermite equation. Afterwards, the numerical analysis of the one-particle harmonic oscillator is given by using the Numerov numerical method. Finally, a corresponding Fortran code for this problem is presented in order to plot the wave function according to a pre-determined quantum number. Plots of the wave function obtained from the Fortran code are also given. In the end, some exercises are given to include other information in the Fortran code.(153words).

DOI
10.1201/9781003050582-17

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This chapter begins discussing the classical harmonic oscillator and its solution as a second-order differential equation of the general type. Next, the one-particle quantum harmonic oscillator is presented, its corresponding Hamiltonian functionand wave equation followed by the undimensionalization of this wave equation yielding a second-order differential equation of the general type which leads to the general Hermite equation and Hermite polinomials as partial solution of the wave function. The energy for each wave function is obtained from the recursion relation of the Hermite equation. Afterwards, the numerical analysis of the one-particle harmonic oscillator is given by using the Numerov numerical method. Finally, a corresponding Fortran code for this problem is presented in order to plot the wave function according to a pre-determined quantum number. Plots of the wave function obtained from the Fortran code are also given. In the end, some exercises are given to include other information in the Fortran code.(153words).
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.
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.
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
Harmonic oscillator model,the physical method of which enjoys wide application in other related courses is the simplest but the most important model in quantum mechanics.On one hand,profound understanding of harmonic oscillator model through various ways is crucial to understand the essence of quantum mechanics and research into micro physical models by means of quantum mechanics;on the other hand,the application of algebraic approach to solve the eigenvalues and wave function of mechanics is the main means to solve the actual problems in quantum mechanics.This paper will take harmonic oscillator as an example,and apply algebraic approach to firstly present wave function of one-dimensional harmonic oscillator,and then multi-dimensional harmonic oscillator,and also combine corresponding commutator relations of operator to present Hermite polynomials and its recursion relations,which will avoid the process of using series to solve the equation of Heimite.Meanwhile,it will improve some shortcomings revealed in the same paper.
Abstract This paper advocates writing the eigenfunctions of the harmonic oscillator without Hermite polynomials, by means of the following recurrence relation: Ψ n = 2 α / n x Ψ n − 1 − ( n − 1 ) / n Ψ n − 2 ( n > 0 ) , with Ψ 0 = ( α / π ) 1 / 4 e − α x 2 / 2 . It is easy to prove that these functions are eigenfunctions of the harmonic oscillator. It is shown that many of the exercises related to the harmonic oscillator found in textbooks can be solved in a simple and elegant way by using this recurrence relation.
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.