Abstract

Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover, the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

中文翻译：

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球坐标中的超病毒定律和埃伦斯特定理：系统方法

摘要

考虑到径向距离区域的受限制特征，考虑了3维量子力学中一些基本关系的阐述。在这种情况下，径向波函数的边界行为和坐标原点处的算符奇异性有助于这些关系。我们推导出算子时间导数的平均值和该算子平均值的时间导数之间的关系，根据定义，这通常被认为是相同的。与已知结果的偏差由额外的项推导和表示，这取决于上述边界行为。这个额外术语的一般形式发生在类似超病毒定理中。在特定情况下，考虑了库仑和振子势的病毒定理，并推导了对克莱默斯求和规则的修正。此外，推导了修正的埃伦菲斯定理，并证明了其与真实物理图像的一致性。