论文标题
修改后的Bertrand定理如何解释元素元素表的规律和异常
How the Modified Bertrand Theorem Explains Regularities and Anomalies of the Periodic Table of Elements
论文作者
论文摘要
Bertrand定理允许在3D欧几里得空间中封闭轨道，仅适用于2种类型的中心电位。这些是开普勒 - 库仑和谐波振荡器类型。沃尔克·佩里克（Volker Perlick）最近设计了新的静态球形对称（Bertrand）空间，遵守爱因斯坦的方程并支撑封闭的轨道。在这项工作中，我们证明了这些空间的拓扑和几何形状使我们可以准确地解决任何周期系统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计算。最初，Tietz于1956年通过Tietz获得了Schrödinger方程的精确解（无引用Bertrand定理）。我们通过应用与新方法（与1936年Fock在1936年开发的方法相同的方法）来重新计算Tietz结果，以解决Schrödinger'sEquation arthogation for Hyderogen for Hyderogen Atom Atom。通过使用这种新方法，可以证明Tietz型Schrödinger的方程实际上正在描述Bertrand SpaceTimes中的量子运动。作为奖励，我们在分析中解决了Löwdin的挑战问题。获得的解决方案并不是普遍的，因为过渡金属以及灯笼和actinides中的Madelung规则例外。量子在机械上，这些例外以及规则本身在迄今为止借助相对论的Hartree-Fock计算对待。获得的结果尚未详细描述异常。但是，本文概述的研究表明，开发的方法也能够描述异常。本文以一些关于原子物理学问题在量子力学，量子场理论和（触电）重力发展的有用性结尾。
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.
