Hydrogen atom in a gravitational field

[Gavroy]

hi

does anybody of you know if there is an equation that describes an atom in a gravitational field of a star or something like that (general relativity or Newton)or do you know some results that could tell me something about the magnitude of this energy corrections?

do you know a method that is used to describe such perturbation?

 

[Vanadium 50]

If you let both the nucleus and electrons fall, the energy levels are unchanged. (The electrons and nucleus' common motion cancels) If you somehow support the nucleus, the corrections to the electron's energy levels use the same methodology as the Stark Effect.

 

[Bill_K]

Gavroy, The energy correction of an atom due to gravity would be incredibly small. Before we start it must be clear what we're not talking about. An atom falls in a gravitational field and this adds to its total energy, but does not affect the atomic energy levels. When you fall into a black hole, for example, it's not the fall that hurts you, it's the tidal gravitational forces. That is, you're stretched in one direction and squashed in the other. Tidal gravitational forces are quadrupole.

This means as far as atomic levels go, the effect of gravity will be quadrupole. It would resemble Stark splitting but with ΔJ = 2. One could write down the selection rules and line patterns, but more important first is to realize what a small effect we're talking about.

Take an extreme example - let the atom be near a one solar mass black hole. The Schwarzschild radius of the sun is 3 x 10⁵ cm. The gravitational potential is GMm/r, the usual gravitational force is GMm/r², but the tidal gravitational force is GMm/r³. And for its effect on an object of diameter d this means ΔE = (GMm/r³) d².

Okay, how do we work out the value. Easiest way is to realize that the gravitational potential energy GMm/r of an object of mass m near the surface of a black hole is roughly mc², and that's most of the factors in our expression. What we're left with is ΔE = mc² (d/r)². Numerically the diameter of an atom is d = 10^-8 cm, the rest mass of an electron is mc² = 0.5 MeV, and as we said for a solar mass black hole, r = 3 x 10⁵ cm.

So ΔE = 0.5 MeV (10^-8 cm/3 x 10⁵ cm)² = about 10^-21 eV for the level shifts. Putting that in terms of frequency, it means a Δν of about 1 Hz.

 

[Gavroy]

@ Bill K

yes, thank you this effect was exactly what i was looking for.

but do you also know how to solve this problem correctly?
i guess that i need some kind of dirac equation in curved spacetime or perturbation theory but i could not find anything at all yet.
so do you know where i could find some information about the theoretical background?