Energy-level shifts of a stationary hydrogen atom in a static external gravitational field with Schwarzschild geometry

The first order perturbations of the energy levels of a stationary hydrogen atom in a static external gravitational field, with Schwarzschild metric, are investigated. The energy shifts are calculated for the relativistic $1S$, $2S$, $2P$, $3S$, $3P$, $3D$, $4S$, $4P$, $4D$, and $4F$ levels. The results show that the energy-level shifts of the states with total angular momentum quantum number $1/2$ are all zero, and the ratio of absolute energy shifts with total angular momentum quantum number $5/2$ is $1:4:5$. This feature can be used to help us to distinguish the gravitational effect from other effects.