Attractive force between potential wells

[LostConjugate]

This is based on my understanding that solving the Schrödinger equation for two potential wells in close proximity has the possibility for a lower energy eigenvalue in the first quantum state then that of a single potential well or two very far away from each other.

Hopefully I have said that correctly.

If this is so, wouldn't there be a conservative (always attractive) force between potential wells?

 

[conway]

"Always" is a difficult proposition, but I would say that's basically the explanation for the bonding force in the hydrogen molecular ion.

 

[LostConjugate]

If you had two ideally neutral atoms would there not remain a conservative force between them due to this quantum mechanical force between the protons which we could consider as two potential wells?

 

[LostConjugate]

Does anyone know this? Know if it can be calculated in one dimension?

 

[SpectraCat]

If you had two ideally neutral atoms would there not remain a conservative force between them due to this quantum mechanical force between the protons which we could consider as two potential wells?

I am not sure what you mean. Taking the example of the molecular hydrogen ion, the force between the nuclei (protons( is always repulsive). For certain geometries, the interaction of the electron with both nuclei results in an ground electronic eigenstate that has an energy that is lower than the ground state energy of an H-atom. Furthermore, this is effect is enough to decrease the total energy of the system (not just the electronic energy), meaning that it more than compensates for the proton-proton repulsion. This, as was stated in an earlier post, is why the hydrogen molecular ion can exist.

Now, if you are asking if this can all be recast in terms of neutral atoms, then I guess the answer can be yes if you apply the condition that the interaction between the atoms is much weaker than the interactions between the electrons and nuclei making up those atoms. In such as case you get dispersion forces, which are always attractive. There are many ways to represent dispersion forces, depending on what one is trying to achieve. One way to think about them is, as the distance between atoms A and B decreases, there is an increasing (although always very small) probability of observing an electron from atom A in the volume of space close to atom B, and vice versa. In other words there is always a very small amount of ionic character (A^{+} + B^{-}) in the interaction between the atoms, resulting in an always attractive force.

Anyway, this sounds like it might be the sort of thing you are looking for, but I am not sure. It involves multiple applications of the BO approximation, and some other assumptions about relative magnitudes of interactions, as well as perfect screening of the interaction between the positive nuclei, so it is far from being a rigorous theory of conservative forces between atoms.

 

[LostConjugate]

Thanks Spectra,

So there is a force besides the electromagnetic force as you explained in the hydrogen ion situation.

What if you had a lot of protons, like a very large number of them all in one spot, well close together. Each one had an electron so the entire assembly was electrically neutral. And you had a neutral Hydrogen Atom far away at distance r.

There should be a force between the huge proton collection potential well and this Hydrogen Atom somewhere r distance away due to what we have spoken of above.

Can it be calculated what the force would be for any given distance r? Where r is far enough way to exclude tidal forces.

 

[SpectraCat]

So there is a force besides the electromagnetic force as you explained in the hydrogen ion situation.

No, that is not really correct. The "force" you are referring to is not something new, it is the result of considering small perturbations to the average electronic wavefunctions of the isolated atoms. Another way of saying this is that it can be represented as the gradient of an interaction potential arising from small fluctuations in the electronic clouds of the atomic systems, which are made up of individual electrons and protons interacting via the EM force. So the only true force in the system is still just the EM force, the rest of it is a high-level approximate treatment.

This may still be what you are looking for, but I want to be as clear as I can about where it comes from.

What if you had a lot of protons, like a very large number of them all in one spot, well close together. Each one had an electron so the entire assembly was electrically neutral. And you had a neutral Hydrogen Atom far away at distance r.

There should be a force between the huge proton collection potential well and this Hydrogen Atom somewhere r distance away due to what we have spoken of above.

Can it be calculated what the force would be for any given distance r? Where r is far enough way to exclude tidal forces.

Well, I wouldn't use H-atoms as you have done, because they are fermions, and thus have a whole other bag full of spin-related phenomena to deal with. However, if you use helium-4 atoms for your particles, then the dispersion forces we have been talking about are the largest terms in the interaction potential.

However, calculating the interaction potential is tricky. To a first approximation, if you know what the force is between two H-atoms as described above, you can represent the force between the isolated H-atom and the collection you describe as a sum over pairwise potentials. However, this is not precise, because higher level interactions between multiple particles come into play.

For example, it is known that the total dispersion forces among a triplet of atoms is *not* simply the sum of pairwise interaction, but that there is a new 3-body interaction that must be considered as well. To make matters worse, this 3-body interaction is not positive definite, like pairwise dispersion forces, and thus will not always increase the interaction energy. The interaction energy is computed as the sum of a (convergent) infinite series of many-body interactions.

Now, in practice, it is usually a good approximation to truncate the series after the pairwise terms, but the results are then not precise. So like I said, this is an approximate method that can be useful, but wouldn't really be suitable as a basis for a "new" conservative force, as you seemed to be seeking in your first post.

 

[LostConjugate]

No, that is not really correct. The "force" you are referring to is not something new, it is the result of considering small perturbations to the average electronic wavefunctions of the isolated atoms. Another way of saying this is that it can be represented as the gradient of an interaction potential arising from small fluctuations in the electronic clouds of the atomic systems, which are made up of individual electrons and protons interacting via the EM force. So the only true force in the system is still just the EM force, the rest of it is a high-level approximate treatment.

This may still be what you are looking for, but I want to be as clear as I can about where it comes from.

Ok I think I understand. The calculations are based off some wave function between the two wells to begin with, so without the electromagnetic force we would have no energy eigenvalues, just two wells.

Its the idea that there is this conservative force between matter that arises from the possibility for lower energy eigenstates. It looks to be a very weak force and a fundamental force. Very very similar to gravity. I imagine someone has calculated it to see how strong it is.