I Gravitational Effect on Electron Eigenstates

[Jim Lundquist]

As a hydrogen atom approaches a Neutron star, is the probability distribution of eigenstates of the electron in that atom influenced by the gravitational field of the star?

 

[Nugatory]

The effect is completely insignificant.

A classical calculation will give you a good sense of how insignificant it is: what is the difference in potential energy between a one-electron mass at a distance $R$ from a gravitating mass, and the same mass at a distance $R+r$ from that mass, where $r$ is the size of an atom? That will be a pretty good approximation of the magnitude of the additional term in the Hamiltonian from the effects of gravity. Compare it with the approximate magnitude of the term from the electromagnetic force between electron and nucleus, which is what we use to calculate the eigenvalue in the standard situation.

 

[Jim Lundquist]

With the risk of belaboring the point, if we substitute a Supermassive Black Hole for that Neutron star, does the same hold true? It seems like the gravitational force would be able to eventually overcome the electromagnetic force between the electron and nucleus as it approaches the singularity.

 

[Nugatory]

It seems like the gravitational force would be able to eventually overcome the electromagnetic force between the electron and nucleus as it approaches the singularity.

Don't guess, calculate!
You'll find that the opposite is true - the altogether negligible effect is even more negligible near a supermassive black hole because the gravitational gradient is smaller.

(Edit: i am interpreting "approach" as "fall towards", as opposed to blasting around in a rocket ship at high accelerations)

 

[Khashishi]

In a white dwarf star near the mass limit, most of the gravity is due to the protons. Above the mass limit, the gravity between protons overcomes the electron degeneracy pressure. The gravity between proton and electrons is still orders of magnitude smaller. So there isn't much direct effect of gravity on the electrons. There's certainly an effect of gravity on protons.

 

[PeterDonis]

In a white dwarf star near the mass limit, most of the gravity is due to the protons.

I assume you mean protons and neutrons, i.e., nuclei. White dwarf matter is not entirely hydrogen.

The gravity between proton and electrons is still orders of magnitude smaller.

In other words, the vast majority of the mass of the white dwarf is nucleons, not electrons. This is true.

So there isn't much direct effect of gravity on the electrons.

But this does not follow from the above. The electrons can basically be viewed as test objects in the gravitational field of the nucleons, and that field does have a significant effect on the electrons.