The gravitational analogue to the hydrogen atom (A summer study at the borders of quantum mechanics and general relativity)

Details The gravitational analogue to the hydrogen atom (A summer study at the borders of quantum mechanics and general relativity) The gravitational analogue to the hydrogen atom (A summer study at the borders of quantum mechanics and general relativity)

2007-03-06

arxiv.org/abs/physics/0703064v2

Eur.J.Phys.28:465-478,2007

Physics

Physics Education

19 pages, 3 figures

Scientific paper

10.1088/0143-0807/28/3/007

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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