A geodesical approach for the harmonic oscillator

Q4 Social Sciences

Revista Mexicana De Fisica E Pub Date : 2020-01-28 DOI:10.31349/revmexfise.17.6

R. Sánchez-Martinez, A. L. Salas-Brito, H. N. Núñez-Yépez

引用次数: 0

Abstract

The harmonic oscillator (HO) is present in all contemporary physics, from elementary classical mechanics to quantum field theory. It is useful in general to exemplify techniques in theoretical physics. In this work, we use a method for solving classical mechanic problems by first transforming them to a free particle form and using the new canonical coordinates to reparametrize its phase space. This technique has been used to solve the one-dimensional hydrogen atom and also to solve for the motion of a particle in a dipolar potential. Using canonical transformations we convert the HO Hamiltonian to a free particle form which becomes trivial to solve. Our approach may be helpful to exemplify how canonical transformations may be used in mechanics. Besides, we expect it will help students to grasp what they mean when it is said that a problem has been transformed into another completely different one. As, for example, when the Kepler problem is transformed into free (geodesic) motion on a spherical surface.

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谐振子的测地线方法

谐振子(HO)存在于所有当代物理学中，从基本经典力学到量子场论。一般来说，举例说明理论物理中的技术是有用的。在这项工作中，我们使用了一种解决经典力学问题的方法，首先将它们转换为自由粒子形式，然后使用新的规范坐标重新参数化其相空间。该技术已被用于求解一维氢原子，也用于求解粒子在偶极势中的运动。使用正则变换，我们将HO哈密顿量转换为自由粒子形式，这变得很容易求解。我们的方法可能有助于举例说明规范变换在力学中的应用。此外，我们希望它能帮助学生理解当一个问题被转化为另一个完全不同的问题时，他们的意思是什么。例如，当开普勒问题转化为球面上的自由(测地线)运动时。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Revista Mexicana De Fisica E 社会科学-科学史与科学哲学

CiteScore

0.80

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The Revista Mexicana de Física (Rev. Mex. Fis.) publishes original papers of interest to our readers from the physical science com unity. Language may be English or Spanish, however, given the nature of our readers, English is recommended. Articles are classified as follows: Research. Articles reporting original results in physical science. Instrumentation. Articles reporting original contributions on design and construction of scientific instruments. They should present new instruments and techniques oriented to physical science problems solutions. They must also report measurements performed with the described instrument. Reviews. Critical surveys of specific physical science topics in which recent published information is analyzed and discussed. They should be accessible to physics graduate students and non specialists, and provide valuable bibliography to the specialist. Comments. Short papers (four pages maximum) that assess critically papers by others authors previously published in the Revista Mexicana de Física. A comment should state clearly to which paper it refers.