修正的Bertrand定理如何解释元素周期表的规律性和反常性
How the Modified Bertrand Theorem Explains Regularities and Anomalies of the Periodic Table of Elements
摘要：
Bertrand定理只允许两种中心势在三维欧氏空间中有闭轨。这些都是开普勒-库仑和谐振子类型。Volker Perlick最近设计了新的静态球对称（Bertrand）时空，它服从爱因斯坦方程并支持闭合轨道。在这项工作中，我们证明了这些时空的拓扑和几何允许我们精确地解决周期系统任何原子的量子多体问题。任何原子的光谱计算都类似于氢原子的光谱计算。最初，提茨在1956年得到了任意多电子原子的薛定谔方程的精确解（不涉及伯特兰定理）。我们重新计算Tietz的结果，通过应用新的方法（不同于Fock在1936年开发的）解决薛定谔方程的氢原子一致。通过使用这种新的方法，证明蒂茨型薛定谔方程实际上描述了伯特兰时空中的量子运动成为可能。作为奖励，我们解析地解决了Löwdin的挑战问题。然而，由于在过渡金属中以及在镧系元素和锕系元素中存在马德隆规则的例外，因此所获得的解决方案并不普遍。量子力学上，这些例外以及规则本身迄今为止都是在相对论哈特里-福克计算的帮助下处理的。所得结果尚未详细描述例外情况。然而，本文中概述的研究表明，发达的方法是能够描述的例外情况。本文最后对原子物理学问题对量子力学、量子场论和（遥平行）引力的发展的有用性作了一些评论。
英文摘要：
Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently designed new static spherically symmetric (Bertrand) spacetimes obeying Einstein's equations and supporting closed orbits. In this work we demonstrate that the topology and geometry of these spacetimes permits us to solve quantum many-body problem for any atom of periodic system exactly. The computations of spectrum for any atom are analogous to that for hydrogen atom. Initially, the exact solution of the Schrödinger equation for any multielectron atom (without reference to Bertrand theorem) was obtained by Tietz in 1956. We recalculated Tietz results by applying the methodology consistent with new (different from that developed by Fock in 1936) way of solving Schrödinger's equation for hydrogen atom. By using this new methodology it had become possible to demonstrate that the Tietz-type Schrödinger's equation is in fact describing the quantum motion in Bertrand spacetimes. As a bonus, we solved analytically the Löwdin's challenge problem. Obtained solution is not universal though since there are exceptions of the Madelung rule in transition metals and among lanthanides and actinides. Quantum mechanically these exceptions as well as the rule itself are treated thus far with help of relativistic Hartree-Fock calculations. The obtained results do not describe the exceptions in detail yet. However, studies outlined in this paper indicate that developed methods are capable of describing exceptions as well. The paper ends with some remarks about usefulness of problems of atomic physics for development of quantum mechanics, quantum field theory and (teleparallel) gravity.
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