Approximate solutions of the transcendental equation for the square quantum wells by finding the real root of the cubic equation.

IF 1.1 4区 物理与天体物理 Q3 PHYSICS, MULTIDISCIPLINARY Canadian Journal of Physics Pub Date : 2023-04-14 DOI:10.1139/cjp-2023-0004
J. Noël, L. Levesque
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引用次数: 0

Abstract

The energy eigenvalues En in finite square quantum wells (SQW) cannot be found using an analytic expression. As a result, numerical methods are normally used to find the eigenvalues from a transcendental equation. In this report, it will be shown that the eigenvalue solution for a given state consists in finding the only real positive root of a depressed trinomial polynomial of third order, which is as easy to solve as a quadratic equation. The method proposed can also be applied for semi-infinite, finite and asymmetric SQW, which are often presented in Quantum Mechanics (QM) textbooks at the undergraduate level. The proposed method can be applied during an exam when programmable calculators are not allowed as the real root of the trinomial polynomial can be found using the formula for the cubic equation found nearly 500 years ago.
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通过求三次方程的实根求平方量子阱超越方程的近似解。
有限平方量子阱(SQW)中的能量特征值En不能用解析表达式求得。因此,通常使用数值方法从超越方程中找到特征值。在本报告中,我们将证明给定状态的特征值解在于找到一个三阶抑制三叉多项式的唯一实正根,它与二次方程一样容易求解。本文提出的方法也适用于本科阶段量子力学教科书中经常出现的半无限、有限和非对称量子力学问题。当考试中不能使用可编程计算器时,可以使用该方法,因为使用近500年前发现的三次方程公式可以找到三叉多项式的实根。
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
Canadian Journal of Physics
Canadian Journal of Physics 物理-物理:综合
CiteScore
2.30
自引率
8.30%
发文量
65
审稿时长
1.7 months
期刊介绍: The Canadian Journal of Physics publishes research articles, rapid communications, and review articles that report significant advances in research in physics, including atomic and molecular physics; condensed matter; elementary particles and fields; nuclear physics; gases, fluid dynamics, and plasmas; electromagnetism and optics; mathematical physics; interdisciplinary, classical, and applied physics; relativity and cosmology; physics education research; statistical mechanics and thermodynamics; quantum physics and quantum computing; gravitation and string theory; biophysics; aeronomy and space physics; and astrophysics.
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