处于中心势的粒子径向Schrödinger方程的可接受解

IF 0.8 4区 教育学 Q3 EDUCATION, SCIENTIFIC DISCIPLINES
J. Etxebarria
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引用次数: 0

摘要

我们重新讨论Schrödinger中心势径向方程中原点处的边界条件。我们给出了一个透明和令人信服的理由,要求波函数的径向部分R(R)在R = 0处是有限的,表明如果R(0)发散,完整波函数ψ不满足完整Schrödinger方程。如果R(R)是奇异的,我们证明相应的ψ遵循一个类似于Schrödinger的方程,但是有一个额外的项涉及狄拉克函数或它在原点的导数。虽然,一般来说,理解我们的一些论点需要一定的分布理论知识,但在R = 0附近的行为R∝1/ R会产生一个可归一化的ψ的重要情况,是特别简单的:狄拉克δ项的起源是通过对通常的球坐标稍加修改而清楚地证明的。这个论点很容易被物理系的本科生理解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Acceptable solutions of the radial Schrödinger equation for a particle in a central potential
We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.
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来源期刊
American Journal of Physics
American Journal of Physics 物理-物理:综合
CiteScore
1.80
自引率
11.10%
发文量
146
审稿时长
3 months
期刊介绍: The mission of the American Journal of Physics (AJP) is to publish articles on the educational and cultural aspects of physics that are useful, interesting, and accessible to a diverse audience of physics students, educators, and researchers. Our audience generally reads outside their specialties to broaden their understanding of physics and to expand and enhance their pedagogical toolkits at the undergraduate and graduate levels.
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