# Gravitational Effect on Electron Eigenstates

• I
• Jim Lundquist
In summary, the probability distribution of eigenstates of the electron in a hydrogen atom is not significantly influenced by the gravitational field of a Neutron star or a Supermassive Black Hole. This is because the magnitude of the additional term in the Hamiltonian is very small compared to the electromagnetic force between the electron and nucleus. Even in a white dwarf star near the mass limit, where most of the gravity is due to the protons, the effect of gravity on the electrons is still negligible. This is because the vast majority of the mass in a white dwarf is made up of nucleons, not electrons.
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?

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.

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.

Jim Lundquist said:
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)

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.

Khashishi said:
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.

Khashishi said:
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.

Khashishi said:
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.

## 1. What is the gravitational effect on electron eigenstates?

The gravitational effect on electron eigenstates refers to the influence of gravity on the energy levels and wave functions of electrons in an atom. This phenomenon is important in understanding the behavior of electrons in extreme gravitational fields, such as those found in black holes or neutron stars.

## 2. How does gravity affect the energy levels of electrons?

Gravity affects the energy levels of electrons by creating a potential energy well that the electrons must overcome in order to escape the gravitational field. This results in a shift in the energy levels, known as gravitational redshift, where the energy of the electron decreases as it moves away from the source of gravity.

## 3. Can gravity change the shape of electron orbitals?

Yes, gravity can change the shape of electron orbitals. In extreme gravitational fields, the orbital shapes can become distorted, leading to changes in the electron's behavior and properties. This effect is known as gravitational lensing and has been observed in celestial objects such as galaxies and black holes.

## 4. How does the strength of gravity affect electron eigenstates?

The strength of gravity directly affects the energy levels of electrons. The stronger the gravitational field, the greater the energy shift in the electron eigenstates. This is why extreme gravitational fields, such as those found in black holes, can have a significant impact on the behavior of electrons.

## 5. What are the implications of the gravitational effect on electron eigenstates?

The gravitational effect on electron eigenstates has implications in various fields of study, including astrophysics and quantum mechanics. It can help us understand the behavior of matter in extreme environments and provide insight into the fundamental nature of gravity and its relationship with other fundamental forces in the universe.

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