Abstract

It is well-known that if the physical system is located in the restricted area, additional “surface terms” emerge in the traditional form of hypervirial and/or Erenfest theorems. In the current literature mainly one-dimensional Schrodinger equation was considered in this respect. Our observation consists in that this situation emerges automatically in spherical coordinates, as well as one of the coordinates, namely radial distance, is restricted by a half-line. In particular, these considerations are clearly manifested, when one consider spherically symmetric potentials and operators, depended only on distance. Evidently, these additional terms give rise owing the boundary conditions for wave functions and the behavior of the considered operators at the origin of coordinates. We have analyzed the role of these additional terms for various model- potentials in the Schrodinger equation. We consider regular, as well as soft-singular potentials and show that the inclusion of these terms is very important for obtaining correct physical results. Some complicated integrals for hypergeometric functions are also derived. The modified virial relations, derived below, is converted into the usual relations, when the additional terms are absent and when present, they give reasonable corrections in correct direction. This fact also provides its legitimacy.

中文翻译：

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修正的超病毒和埃伦费斯特定理的应用及其若干后果

摘要

众所周知，如果物理系统位于受限区域中，则会以传统形式的超病毒定理和/或埃伦费斯特定理出现额外的“表面项”。在当前文献中，在这方面主要考虑一维薛定inger方程。我们的观察结果在于，这种情况自动出现在球坐标系中，并且其中一个坐标系（即径向距离）受半线限制。特别是，当仅考虑距离时，考虑球形对称的势和算子时，这些考虑因素就可以清楚地体现出来。显然，由于波动函数的边界条件和所考虑的算子在坐标原点处的行为，这些附加项会产生。我们已经分析了这些附加项对于Schrodinger方程中各种模型势的作用。我们考虑了规则的以及软奇异的电势，并表明包含这些术语对于获得正确的物理结果非常重要。还推导了一些用于超几何函数的复杂积分。下面衍生的修改后的病毒关系将转换为通常的关系，当缺少附加项时，如果存在附加项，它们会在正确的方向上给出合理的更正。这一事实也提供了其合法性。下面衍生的修改后的病毒关系将转换为通常的关系，当缺少附加项时，如果存在附加项，它们会在正确的方向上给出合理的更正。这一事实也提供了其合法性。下面衍生的修改后的病毒关系将转换为通常的关系，当缺少附加项时，如果存在附加项，它们会在正确的方向上给出合理的更正。这一事实也提供了其合法性。