A real-valued description of quantum mechanics with Schrödinger's 4th-order matter-wave equation

Q2 Physics and Astronomy
Nicos Makris , Gary F. Dargush
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引用次数: 0

Abstract

Using a variational formulation, we show that Schrödinger's 4th-order, real-valued matter-wave equation which involves the spatial derivatives of the potential V(r), produces the precise eigenvalues of Schrödinger's 2nd-order, complex-valued matter-wave equation together with an equal number of negative, mirror eigenvalues. The variational forms of the matter-wave equations are computed numerically with a Ritz-spline method and we show how this method handles accurately discontinuous potentials with singular derivatives. Accordingly, the paper concludes that there is a real-valued description of non-relativistic quantum mechanics in association with the existence of negative, mirror energy levels. Schrödinger's classical 2nd-order, complex-valued matter-wave equation which was constructed upon factoring the 4th-order, real-valued differential operator and retaining only one of the two conjugate complex operators is a simpler description of the matter-wave, since it does not involve the derivatives of the potential V(r), at the expense of missing the negative, mirror energy levels.
用Schrödinger的四阶物质-波动方程描述量子力学的实值
使用变分公式,我们表明Schrödinger的四阶实值物质-波方程(涉及势V(r)的空间导数)产生Schrödinger的二阶复值物质-波方程的精确特征值以及相同数量的负镜像特征值。用ritz样条法对物质波方程的变分形式进行了数值计算,并展示了该方法如何精确地处理具有奇异导数的不连续势。因此,本文得出结论,与负镜像能级的存在相关的非相对论量子力学存在实值描述。Schrödinger的经典二阶复值物质波方程是在分解四阶实值微分算子的基础上构建的,只保留两个共轭复算子中的一个,这是对物质波的更简单描述,因为它不涉及潜在V(r)的导数,以牺牲负镜像能级为代价。
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
Physics Open
Physics Open Physics and Astronomy-Physics and Astronomy (all)
CiteScore
3.20
自引率
0.00%
发文量
19
审稿时长
9 weeks
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