氢原子的引力类似物（量子力学和广义相对论边界的夏季研究）
The gravitational analogue to the hydrogen atom (A summer study at the borders of quantum mechanics and general relativity)
摘要：
本文报道了2006年在法兰克福大学进行的一个学生暑期项目。它是针对熟悉相对论量子力学和广义相对论基本原理的本科生。该项目的目的是研究弯曲时空中的狄拉克方程。为了得到广义相对论狄拉克方程，我们在第一部分中使用引力作为规范理论的表述。在这些一般的考虑之后，我们将进一步的讨论限制在史瓦西度规的特殊情况。这种设置对应于氢原子，其中电磁场被重力取代。虽然在事件视界有一个奇点，但事实证明，与时间无关的狄拉克方程存在一个正则解。最后，利用适当的边界条件对狄拉克方程进行数值求解.
英文摘要：
This article reports on a student summer project performed in 2006 at the University of Frankfurt. It is addressed to undergraduate students familiar with the basic principles of relativistic quantum mechanics and general relativity. The aim of the project was to study the Dirac equation in curved space time. To obtain the general relativistic Dirac equation we use the formulation of gravity as a gauge theory in the first part. After these general considerations we restrict the further discussion to the special case of the Schwarzschild metric. This setting corresponds to the hydrogen atom, with the electromagnetic field replaced by gravity. Although there is a singularity at the event horizon it turns out that a regular solution of the time independent Dirac equation exists. Finally the Dirac equation is solved numerically using suitable boundary conditions.
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