Attractive force between potential wells

In summary, the conversation discussed the possibility of a lower energy eigenvalue in the first quantum state for two potential wells in close proximity compared to a single potential well or two potential wells far apart. This effect is known as dispersion forces and can be observed in systems like the hydrogen molecular ion. However, calculating the exact force between a large collection of protons and a neutral hydrogen atom at a certain distance is challenging due to higher level interactions between multiple particles.
  • #1
LostConjugate
850
3
This is based on my understanding that solving the Schrodinger 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?
 
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  • #2
"Always" is a difficult proposition, but I would say that's basically the explanation for the bonding force in the hydrogen molecular ion.
 
  • #3
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?
 
  • #4
Does anyone know this? Know if it can be calculated in one dimension?
 
  • #5
LostConjugate said:
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 ([tex]A^{+} + B^{-}[/tex]) 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.
 
  • #6
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.
 
  • #7
LostConjugate said:
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.

LostConjugate said:
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.
 
  • #8
SpectraCat said:
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.
 

What is the concept of attractive force between potential wells?

The concept of attractive force between potential wells is based on the idea that two or more particles can be attracted to each other due to a potential energy well, which is a region in space where the potential energy is lower compared to the surrounding areas. The particles will experience a force that pulls them towards this potential energy well, causing them to move closer together.

What factors affect the strength of the attractive force between potential wells?

The strength of the attractive force between potential wells depends on several factors, including the distance between the particles, the masses of the particles, and the shape of the potential well. The closer the particles are to each other, the stronger the force will be. Similarly, particles with larger masses will experience a stronger force compared to particles with smaller masses. The shape of the potential well also plays a role, as a deeper and narrower well will result in a stronger force compared to a shallower and wider well.

How does the attractive force between potential wells contribute to particle bonding?

The attractive force between potential wells is a crucial factor in particle bonding. When two or more particles are attracted to each other due to a potential energy well, they can form bonds by moving closer together and reducing the potential energy between them. This bond can be temporary or permanent, depending on the strength of the attractive force and the stability of the potential well.

Can the attractive force between potential wells be repulsive?

Yes, in some cases, the attractive force between potential wells can become repulsive. This can happen when the particles are too close to each other, and the potential energy well becomes too narrow and steep. In this scenario, the particles will experience a force that pushes them away from each other, resulting in repulsion rather than attraction.

What are some real-life examples of attractive force between potential wells?

The attractive force between potential wells is a fundamental concept in physics and can be observed in various real-life phenomena. Some examples include the force between atoms in a molecule, the gravitational force between celestial bodies, and the force between protons and neutrons in an atomic nucleus. In all of these cases, the particles are attracted to each other due to a potential energy well, which leads to the formation of bonds or the stability of the system.

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