Potential terms in SE & Quantum Chemistry

[mieral]

I bought this book "Idiot's Guide to Quantum Physics" I bought it because I know I am an idiot in quantum physics. Now I have a question about the contents. It says that:

"The four fundamental forces in nature (gravity, electromagnetism, plus two more that we'll formally introduce in Chapter 16) are just the sort of forces which can be described by potential energy functions. Any apparent "force" that we feel in our everyday lives, is derived from one of these four..."
"The four fundamental forces obey the law of conservation of energy we talked about in Chapter 2, and they vary with position. The gravitational force, for example, gets weaker in a predictable way as the separation between two masses is increased. The electromagnetic force gets stronger as a magnet gets closer to a piece of steel. If you've ever stuck a piece of preschool artwork to your refrigerator, you have direct experience with this. These are the qualities that make the potential energy U an excellent stand-in for forces in the Schrodinger equation."

My question is.. in spectroscopy or quantum chemistry or solid state physics or any molecular dynamics.. is the potential energy U always related to the four fundamental forces or could it occur from other molecular dynamics.. and what kinds of molecular dynamics that doesn't involve the four fundamental forces?

My last question.. in quantum chemistry and spectroscopy.. is it all about getting the "potential" terms to solve for the Schrodinger equations. For complex molecular dynamics.. how do you determine the potential of all the combined stuff?

Thanks.

 

[PeterDonis]

is the potential energy U always related to the four fundamental forces or could it occur from other molecular dynamics.. and what kinds of molecular dynamics that doesn't involve the four fundamental forces?

Do you understand what "four fundamental forces" means? It means every interaction is "built" out of them. There are no "dynamics" that don't involve them.

For complex molecular dynamics.. how do you determine the potential of all the combined stuff?

From the fundamental forces that are involved. For atoms and molecules, this will be the electromagnetic force; the others don't come into play in that context.

 

[mieral]

Do you understand what "four fundamental forces" means? It means every interaction is "built" out of them. There are no "dynamics" that don't involve them.
From the fundamental forces that are involved. For atoms and molecules, this will be the electromagnetic force; the others don't come into play in that context.

Electromagnetic force is the most practical force in daily physics.

For intermolecular forces and bonding.. I mean do we really analyze the contribution of each to potential energy of the Hamiltonian operator? For excited and ground states of molecules.. do we use the potential energy technique and is this sufficient or powerful enough?

Also what applications in molecules that we need to take into account the momentum operator.. besides only the Hamiltonian (total energy) operator?

 

[PeterDonis]

For intermolecular forces and bonding.. I mean do we really analyze the contribution of each to potential energy of the Hamiltonian operator?

Contribution of each what? The only force that is significant is the electromagnetic force.

For excited and ground states of molecules.. do we use the potential energy technique

You marked this thread as "I" level. Have you looked at a textbook on QM?

 

[mieral]

Contribution of each what? The only force that is significant is the electromagnetic force.

Contributions from the rotational, translational or vibration modes.. do we compute for the potential of each and add them? Can't we use the Lagrangian for them?

You marked this thread as "I" level. Have you looked at a textbook on QM?

I'm reading Ballentine's Quantum Mechanics: A Modern Development. It doesn't directly answer the simple question why we need to compute for the momentum operator in molecules.. what do we accomplish by the momentum operator.. I only understand the Hamiltonian operator part.

 

[PeterDonis]

Contributions from the rotational, translational or vibration modes.. do we compute for the potential of each and add them? Can't we use the Lagrangian for them?

It doesn't appear that you understand the basics of QM, even though you say you are reading a QM textbook. This question is too confused for me to try to answer it.

I'm reading Ballentine's Quantum Mechanics: A Modern Development. It doesn't directly answer the simple question why we need to compute for the momentum operator in molecules

Perhaps that's because it isn't as simple a question as you think it is.

If you want a treatment of how QM is used to model atoms and molecules, the Feynman lectures go into it (they are available online at the Caltech website). There are other treatments as well, but the Feynman one is the one I'm reasonably familiar with.

 

[PeterDonis]

It doesn't directly answer the simple question why we need to compute for the momentum operator in molecules

Or perhaps, instead of not being as simple a question as you think it is, as I suggested in my last post, it's even simpler than you think: the momentum operator (in the non-relativistic case, anyway) is ##- i \hbar \nabla##. Why wouldn't this be just as true for a molecule as for any other system?

 

[blue_leaf77]

Contributions from the rotational, translational or vibration modes.. do we compute for the potential of each and add them? Can't we use the Lagrangian for them?

In the discussion of molecular quantum mechanics one often assumes the so-called Born-Oppenheimer approximation where the electronic energy is treated as a virtual potential in which the nuclei undergo the dynamics such as vibration and rotation. Thus, it's actually the motion of the electrons and not nuclei which is treated as a potential. You do not normally associate the vibrational or rotational motion of the nuclei to any potential, instead these motions are the result of assuming the electron's energy as a 'static' potential for the nuclei to do their dynamics.
I agree with Peter that if you are mainly interested in the application of quantum mechanics for molecules (or atoms) you should consult books which are devoted for these topics, Ballentine's book is more of a general foundation of quantum mechanics.

 

[bhobba]

It doesn't directly answer the simple question why we need to compute for the momentum operator in molecules.. what do we accomplish by the momentum operator.. I only understand the Hamiltonian operator part.

Did you read chapter 3? See equation 3.60. Notice the P in the equation?

As the whole chapter explains symmetry considerations, to be specific the belief that the probabilities from the Born rule (really you are invoking the POR, but its just so damn obvious you tend to forget it) constrains the equations of motion.

You also need to read section 3.6 (not the equation the section) on quantisizing a classical system. There are issues involved in that, that really need its own thread. Its actually rather important pedagogically, but for some reason doesn't get discussed much. Its to do with the fact classically Hamiltonian's that contain multiplications of quantities the order doesn't matter - but in QM its crucial.

Thanks
Bill

 

[bhobba]

Or perhaps, instead of not being as simple a question as you think it is, as I suggested in my last post, it's even simpler than you think: the momentum operator (in the non-relativistic case, anyway) is ##- i \hbar \nabla##. Why wouldn't this be just as true for a molecule as for any other system?

My vote goes for that. I think all the OP needs is a refresher on chapter 3 of Ballentine since that is what he is studying - and of course its a good choice of text.

Thanks
Bill

 

[bhobba]

Or perhaps, instead of not being as simple a question as you think it is, as I suggested in my last post, it's even simpler than you think: the momentum operator (in the non-relativistic case, anyway) is ##- i \hbar \nabla##. Why wouldn't this be just as true for a molecule as for any other system?

Of course the derivation of the form of the Hamiltonian as found in chapter 3 applies to anything - atoms, molecules - anything. All it doesn't do is take into account spin - but that's not too hard. But internal degrees of freedom - well that's a bit of a minefield for molecules rather obviously.

Thanks
Bill

 

[bhobba]

I agree with Peter that if you are mainly interested in the application of quantum mechanics for molecules (or atoms) you should consult books which are devoted for these topics, Ballentine's book is more of a general foundation of quantum mechanics.

Indeed.

Ballentine derives the actual equations - solving it for 'complex' systems beyond say the simple hydrogen atom is, how to put it, a LOT more difficult.

Just reading the book Turing's Cathedral right now. This was one of the first applications early computers were put to. Even the simplest calculations before took teams of people years carrying out manual computations - and the chance of error in that - well is obvious. But the politics around these early machines was 'mind boggling'. It was only Von-Neumann's reputation and intervention that kept the different groups ie engineers and theoreticians from totally breaking down. When he died things got a bit hairy for a while.

For early crystallography data it was a nightmare and without the emerging computer impossible by any direct method - it was more trial and error.

Thanks
Bill

 

[mieral]

Of course the derivation of the form of the Hamiltonian as found in chapter 3 applies to anything - atoms, molecules - anything. All it doesn't do is take into account spin - but that's not too hard. But internal degrees of freedom - well that's a bit of a minefield for molecules rather obviously.

Thanks
Bill

If the Hamiltonian produces the basic Schrodinger Equation. How did Schrodinger take into account spin which you say can't be taken into account by the Hamiltonian? Please answer using your own words and not let me read other entire whole books. Thanks.

 

[bhobba]

If the Hamiltonian produces the basic Schrodinger Equation. How did Schrodinger take into account spin which you say can't be taken into account by the Hamiltonian? Please answer using your own words and not let me read other entire whole books. Thanks.

Schrodinger got the right answer but his reasoning was wrong:
https://arxiv.org/abs/1204.0653

It wasn't Schrodinger that nutted out spin - it was Pauli.

Nowadays it's all derived from symmetry. If understanding this whole thing at that level is your won't then get the following:
https://www.amazon.com/dp/3319192000/?tag=pfamazon01-20

But it has nothing to do with solving problems in quantum chemistry - its understanding basic principles.

Physics has great scope and getting across it all is pretty much impossible. I am mostly interested in understanding basic principles, but you really have made a difficult task for yourself if you want to do both.

For quantum chemistry just simply accept that all this stuff it uses can be justified, often from quite elementary symmetry considerations such as the frame invarience of the Born Rule, and unless you want to take a big sojourn into getting across it leave it at that. I fell into that trap myself. I wanted to get across the bra-ket formalism of Dirac and his use of that damnable Dirac Delta function that even Von-Neumann savaged. I did, but it was a long sojourn into advanced areas of functional analysis called Rigged Hilbert Spaces - it slowed down my actual understanding of QM, but I did come out the other end with some rather interesting advanced math that these days is used in many areas eg white noise theory. But it didn't really help in understanding QM.

Thanks
Bill

 

[PeterDonis]

Please answer using your own words and not let me read other entire whole books.

The questions you are asking are too broad in scope for this request to be reasonable. You are basically asking for someone to condense all of QM and its application to atoms and molecules into a PF post. That's not going to happen. You need to work through the textbooks yourself--one textbook alone won't tell you all you need to know. Ballentine is a good start, but as @bhobba has already pointed out, when you're finished with it you won't have answers to all the questions you want to ask. No single textbook will do that.