Quantum Model of Contact Forces

[greswd]

I'm curious about contact forces, such as the normal reaction force on a book resting on a table.

I've read that it's actually the electromagnetic force, but atoms are electrically neutral and there is no ostensible repulsive force between electrons and the nucleus, so I have some doubts.

Has there been a QM derivation of how the normal force comes into being?

 

[greswd]

I mean, its one of the most ubiquitous forces ever, surely there must be some literature on how it forms from the atomic, quantum scale.

 

[Nugatory]

Classical electrostatics handles this problem just fine. The atom as a whole is electrically neutral but the electrons are on the outside, so are closer to one another than they are to the positive-charged nuclei. Thus, the repulsive force between the electrons exceeds the atrractive force between the electrons and the nuclei of the other atom.

 

[greswd]

Classical electrostatics handles this problem just fine. The atom as a whole is electrically neutral but the electrons are on the outside, so are closer to one another than they are to the positive-charged nuclei. Thus, the repulsive force between the electrons exceeds the attractive force between the electrons and the nuclei of the other atom.

The atom is not a solid ball, how does an electron on the edge of the atom repel the nucleus?

Its also odd that I'm using a solid ball analogy when I'm trying to understand why things are solid in the first place. I would like to add, has anyone derived Hooke's Law from atomic considerations?

 

[greswd]

I'm also interested in any scientific papers on this topic. Not that I fully understand them, but they might contain critical concepts.

 

[nasu]

I would like to add, has anyone derived Hooke's Law from atomic considerations?

Sure. It is a straightforward exercise in introductory solid state physics. In the equilibrium state of a crystal the net force on atoms (or ions) is zero.
Small displacement from equilibrium results in restoring force proportional to the displacement. You can even find a relationship between the macroscopic Young's modulus and the microscopic inter-atomic potential.

 

[greswd]

Sure. It is a straightforward exercise in introductory solid state physics. In the equilibrium state of a crystal the net force on atoms (or ions) is zero.
Small displacement from equilibrium results in restoring force proportional to the displacement. You can even find a relationship between the macroscopic Young's modulus and the microscopic inter-atomic potential.

Can you provide any scientific paper/textbook/article that does this derivation?

 

[sophiecentaur]

Can you provide any scientific paper/textbook/article that does this derivation?

My old copy of Kittel 'Solid State Physics' could help. But, if you haven't already done a course on or already have some resources, it may just be running before you can walk. I would certainly think twice before leaping into the middle of solid state Physics. There a very few short cuts in advanced Physics.

 

[greswd]

My old copy of Kittel 'Solid State Physics' could help.

Does he actually derive Hooke's Law from atomic interactions, from the ground up?

Or the normal contact reaction force?

Anything about macroscopic mechanical forces that we are familiar with in everyday life.

 

[sophiecentaur]

Does he actually derive Hooke's Law from atomic interactions, from the ground up?

Or the normal contact reaction force?

Anything about macroscopic mechanical forces that we are familiar with in everyday life.

He doesn't have a single page, deriving just what the OP seems to want. It's hard and that's why it's part of a big book, full of relevant ideas.

 

[greswd]

He doesn't have a single page, deriving just what the OP seems to want. It's hard and that's why it's part of a big book, full of relevant ideas.

Indeed so.

It's funny, Hooke's Law is one of the basics of dynamics, you would've thought that someone would've done this derivation by now.

 

[sophiecentaur]

Like I said, Hooke's Law is just a law of proportionality and it operates over a limited range for any material. If you take a diatomic molecule, the force to alter the location of the two atoms from their equilibrium position will be proportional to the displacement over 'some' range of positions. I would have said that it's too trivial to derive, all on its own, and it would just be noted in passing - as Kittel does in his book. By the time you're at that level, Hooke's law is not particularly interesting in itself.

 

[greswd]

I would have said that it's too trivial to derive, all on its own, and it would just be noted in passing. By the time you're at that level, Hooke's law is not particularly interesting in itself.

My opinion is the opposite of yours.

Hooke's Law is very well known, and I think that deriving Hooke's Law using the knowledge of "that level" is very interesting indeed.

It relates the advanced solid state physics to macroscopic, everyday experiences, and returns to basic formulae. It's like going full circle.

 

[sophiecentaur]

deriving Hooke's Law using the knowledge of "that level" is very interesting indeed.

No more interesting than all the other relationships between quantities that are linear over a certain range. There are 'Potential Wells' all over Physics, in which there is an equilibrium state and in which energy needs to be supplied in order to depart from that equilibrium condition.
Given the massive excitement that Solid State Physics presents Scientist with, there is not a lot of potential interest in re-visiting a relationship that was established hundreds of years before; after all, there are really no surprises there. The fact is that Hooke's Law is 'in agreement' with more advanced behaviours of materials may be surprising for someone who isn't coming from the direction of a SS Physics course but, as I remember it, was just a verbal footnote and a 'friendly face' amongst the more turgid and complicated bits of book work that we came across. Life is chock full of 'A=B times C' approximations in the behaviour of things. Of course, the advanced models of materials would have to check against measurements and following Hooke's Law would be one of those checks - same as the need for any model of conduction in metals would have to follow Ohm's Law.

 

[greswd]

Of course, the advanced models of materials would have to check against measurements and following Hooke's Law would be one of those checks - same as the need for any model of conduction in metals would have to follow Ohm's Law.

Students do learn about how the simple Drude model gives rise to Ohm's Law.

In order for the model to follow a check, wouldn't Hooke's Law have to be derived from it?

 

[sophiecentaur]

how does an electron on the edge of the atom repel the nucleus?

It doesn't. It is repelled by the negative charges that are between it and the positive nucleus. It sits in a potential well and requires energy to bring it closer of further away from the nucleus, even in a classical model. But the situation does not correspond to what happens with a simple pair of atoms. There is a continuous band of energies in a solid whereas there are discrete energy levels for a single molecule.

 

[greswd]

It doesn't. It is repelled by the negative charges that are between it and the positive nucleus. It sits in a potential well and requires energy to bring it closer of further away from the nucleus, even in a classical model. But the situation does not correspond to what happens with a simple pair of atoms. There is a continuous band of energies in a solid whereas there are discrete energy levels for a single molecule.

I see. I can't find any derivation for this either, I guess it is too trivial, as you mentioned earlier.

 

[sophiecentaur]

In order for the model to follow a check, wouldn't Hooke's Law have to be derived from it?

The modulus of a material would be what counts and, over a range of deformations, it would be constant. I can't be bothered to scan the paragraph in Kittel but he quotes Wigner and Seitz (1955)with the idea that, given enough computing power, you could solve the Schrodinger equation for a metal, you could find it's physical constants. But, they say, "Presumably the results would agree with the experimentally determined quantities and nothing vastly new would be learned from the calculations."
That is the point I have been trying to make and I am, it turns out, supported by three of the Greats in SS Physics.

 

[sophiecentaur]

I see. I can't find any derivation for this either, I guess it is too trivial, as you mentioned earlier.

You would need to look at early textbooks on the subject but, from my only source, it seems that it could be found only with the sort of computing power that we have today (not in 1962) and it's probably water under the bridge for anyone, even with the coding ability and the computer access. After all, they aren't looking to disprove Hooke, are they? (All the young blades would be up for something like that if it were likely, I'm sure.)

 

[greswd]

You would need to look at early textbooks on the subject but, from my only source, it seems that it could be found only with the sort of computing power that we have today (not in 1962) and it's probably water under the bridge for anyone, even with the coding ability and the computer access. After all, they aren't looking to disprove Hooke, are they? (All the young blades would be up for something like that if it were likely, I'm sure.)

Maybe its more about curiosity rather than trying to disprove something.

 

[greswd]

The modulus of a material would be what counts and, over a range of deformations, it would be constant.

Do you have a rough memory of where this appears in Kittel?

 

[greswd]

The modulus of a material would be what counts and, over a range of deformations, it would be constant. I can't be bothered to scan the paragraph in Kittel but he quotes Wigner and Seitz (1955)with the idea that, given enough computing power, you could solve the Schrodinger equation for a metal, you could find it's physical constants. But, they say, "Presumably the results would agree with the experimentally determined quantities and nothing vastly new would be learned from the calculations."
That is the point I have been trying to make and I am, it turns out, supported by three of the Greats in SS Physics.

I don't want to accuse you of appealing to authority but it sure sounds like it lol. But nevermind.

Anyway, the remainder of the quote is

It would be preferable instead to have a vivid picture of the behavior of the wave functions, a simple description of the essence of the factors which determine cohesion and an understanding of the origins of variation in properties from metal to metal.

It appears that they had advocated for a first principles approach rather than a brute force computation.

 

[sophiecentaur]

I don't want to accuse you of appealing to authority but it sure sounds like it lol. But nevermind.

I was just saying that the opinion of the 'experts' accounts for why they didn't bother. It's just like I suggested in the first place. Appealing to authority or using evidence - no sure which it was.

It appears that they had advocated for a first principles approach

But did they bother?

 

[greswd]

I was just saying that the opinion of the 'experts' accounts for why they didn't bother. It's just like I suggested in the first place. Appealing to authority or using evidence - no sure which it was.

But did they bother?

You did say "three of the Greats in SS Physics." But nevermind, its a legit quote anyway right?

I don't know, maybe they did? They did want a vivid picture.

 

[greswd]

Anyway I'm looking for text on the part of the modulus being constant. Does Kittel cover that part and explain why at the atomic level?

 

[sophiecentaur]

Anyway I'm looking for text on the part of the modulus being constant. Does Kittel cover that part and explain why at the atomic level?

You have to remember that almost any classical relationship is going to be linear over a certain range so it is hardly a surprise that Hooke's Law is followed over 'some' range of strain values. The response of a solid to deformation is going to be linear, simply for that reason. QM has to apply in a solid (and everywhere, I guess) but the quantum steps are spread out into bands in a solid because of Pauli (which seems to apply to all fermions). You are never going to find an 'explanation' of stress/strain in simple QM terms. Solid State Physics acknowledges that and has a different way of describing phenomena, based on statistics and energy bands. That's not a cop out; there is nothing about basic QM that claims to explain everything in the same way as the Hydrogen atom is explained. Afaik, the Hydrogen atom is the only thing that can be solved in such a simple way. Even isolated molecules can't be 'solved' analytically. Kittel is pretty ancient by many standards but I don't think the ideas have been superseded. (? Any other views? )
PS The Drude Model ( referring to a previous post) is not a rigorous QM model for resistivity, is it?