Pauli exclusion principle and Hermitian operators

[jlcd]

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html
"Postulate 2. To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. "

"Postulate 6. The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one fermion with those of another. Electronic spin must be included in this set of coordinates. The Pauli exclusion principle is a direct result of this antisymmetry principle."

What hermitian operator or observable does the pauli exclusion principle fall under? Is it position?

If nature had no position basis. Does it mean there was also no pauli exclusion principle?

One may wonder why I asked this. This is just to have versatile understanding of it from different points of views to get better handle of the concepts.

 

[PeterDonis]

What hermitian operator or observable does the pauli exclusion principle fall under?

None. Read what it says. It's a postulate about the wave function, not about operators.

If nature had no position basis. Does it mean there was also no pauli exclusion principle?

What does "if nature had no position basis" even mean? There is no such theory, so your question is meaningless.

 

[jlcd]

None. Read what it says. It's a postulate about the wave function, not about operators.

Background:

https://www.quora.com/Is-the-wave-f...em-different-from-a-state-vector-for-the-same
"The wave function and state vector are different. The state vector is an abstract vector and when you expand such a vector in a basis (for example: position or momentum), the components or the coefficients of such basis vectors, say in position/momentum, are called the position or momentum wave function."

Paul exclusion principle is always mentioned along with wave function. Hence one may think it's related to the position/momentum basis.

So Pauli exclusion principle as a law of physics is independent of the wave function representation and can be made to appear or not in any HIlbert space representation (with and without the wave function)?

What does "if nature had no position basis" even mean? There is no such theory, so your question is meaningless.

 

[PeterDonis]

Background

None of what this says has anything to do with the question you're asking. You could just as easily state postulate #6 from your OP in terms of the state vector instead of the wave function. And my statement in post #2 would still be just as true.

Paul exclusion principle is always mentioned along with wave function.

No, it isn't; see my statements above. You might not be aware of these things because you haven't spent enough time looking at actual QM textbooks or peer-reviewed papers. That's an issue with your level of knowledge that you should spend some time fixing, since it is apparently leading you to incorrect beliefs about QM.

Hence one may think it's related to the position/momentum basis.

If one thinks that, one is thinking incorrectly.

So Pauli exclusion principle as a law of physics is independent of the wave function representation and can be made to appear or not in any HIlbert space representation (with and without the wave function)?

Yes. And my statements remain true regardless of representation.

 

[PeterDonis]

Pauli exclusion principle as a law of physics is independent of the wave function representation and can be made to appear or not in any HIlbert space representation (with and without the wave function)?

For a simple example, consider the two electrons in a Helium atom in the ground state. They are both in the same orbital, the 1s orbital, so the only difference between them is their spins. And because of the Pauli exclusion principle, their spins must be opposite. But there is no way to write that down in terms of their position/momentum space wave functions, because that part of the wave function is the same for both of them.

 

[jlcd]

For a simple example, consider the two electrons in a Helium atom in the ground state. They are both in the same orbital, the 1s orbital, so the only difference between them is their spins. And because of the Pauli exclusion principle, their spins must be opposite. But there is no way to write that down in terms of their position/momentum space wave functions, because that part of the wave function is the same for both of them.

Can you give a more complex example of object (molecular) with pauli exclusion principle but you can't write it down in terms of their position/momentum space wave function?

 

[PeterDonis]

Can you give a more complex example

You should be able to come up with plenty of examples for atoms heavier than helium in their ground states that have two electrons in an orbital. Any such case will work the same as the two electrons in the helium atom.

If you want a molecule, consider the hydrogen molecule, H₂. The two electrons in this molecule have identical position space wave functions, so, just as with the helium atom, the antisymmetry is in their spins (which must be opposite).

 

[jlcd]

You should be able to come up with plenty of examples for atoms heavier than helium in their ground states that have two electrons in an orbital. Any such case will work the same as the two electrons in the helium atom.

If you want a molecule, consider the hydrogen molecule, H₂. The two electrons in this molecule have identical position space wave functions, so, just as with the helium atom, the antisymmetry is in their spins (which must be opposite).

Say. The state vectors can even hold the information of your bank account. It's just arranging and accounting of information. So do they put the Pauli exclusion principle accounting of spins in the state vectors? Which of the dynamics of Pauli exclusion prinicple do they add or integrate in the state vectors?

 

[PeterDonis]

The state vectors can even hold the information of your bank account. It's just arranging and accounting of information.

I have no idea what you're talking about here.

do they put the Pauli exclusion principle accounting of spins in the state vectors?

I don't know what you mean by "put the accounting in the state vectors". You don't get to just arbitrarily pick how to describe a quantum system with state vectors. You figure it out based on the physical properties of the system. If those properties include the Pauli exclusion principle, then of course the state vectors you come up with are going to reflect that.

 

[PeterDonis]

Which of the dynamics of Pauli exclusion prinicple do they add or integrate in the state vectors?

State vectors by themselves don't describe the dynamics of a quantum system. The Schrodinger Equation (at least in the non-relativistic case) does.

Have you actually tried to work through a textbook that covers this material, and tried to solve some of the exercises? If you haven't, you really need to. You can't learn QM by asking abstract questions. You need to work through concrete examples.

 

[jlcd]

I have no idea what you're talking about here.
I don't know what you mean by "put the accounting in the state vectors". You don't get to just arbitrarily pick how to describe a quantum system with state vectors. You figure it out based on the physical properties of the system. If those properties include the Pauli exclusion principle, then of course the state vectors you come up with are going to reflect that.

Intermission.

I asked this in physics stack exchange:

https://physics.stackexchange.com/questions/477087/can-all-fields-contribute-to-the-potential-energy-that-appears-in-qm-hamiltonian?noredirect=1#comment1071397_477087

"Can all fields contribute to the potential energy that appears in QM Hamiltonian?

"Most importantly: can that potential energy in the QM Hamiltonian able to describe the motion of a single particle in an external electromagnetic field? "

Only one answered and only to the first question and no one answered to the second question. There are 50 or so questions per day so people lost tract of it. About the second question. I was having this exchange with Neumaier.

Me asked: "But still the kinetic and potential energies (Hamiltonian) of particles can be affected by any new forces of nature."

Neumaier: "This is a very simplified description where scalar quantum fields are replaced by external classical fields. It cannot even describe the motion of a single particle in an external classical electromagnetic field"

Me: "Ok. So QM stand powerless. "

Neumaier: "Mainly, because you insist on Hamiltonians consisting of kinetic and potential energy terms. Real Hamiltonians may be very complicated compared to this."

Me: "Of course I was not talking of the classical Hamiltonian but the one used in quantum mechanics"

Neumaier: "So was I".

It's so confusing. What did Neumaier meant by "This is a very simplified description where scalar quantum fields are replaced by external classical fields. It cannot even describe the motion of a single particle in an external classical electromagnetic field".

We were both talking about the quantum Hamiltonian. Here all fields contribute to the potential energy that appears in QM Hamiltonian. Why did he say it's was very simplified description where scalar quantum fields are replaced by external classical fields. What external classical fields? And is it not the quantum Hamiltonian can describe the motion of a single particle in an external classical electromagnetic field?

Before Id spend months or years studying QM. I just need answer to the above questions first so I'd be motivated to study QM textbooks in more details. I get sleepy whenever I opened the book and no specific details I'm looking for. Thanks.

 

[PeterDonis]

Only one answered and only to the first question

His answer, "yes", applies to both questions.

About the second question. I was having this exchange with Neumaier.

Where? Here at PF? Please give a link. I can't usefully comment on the exchange without more context.

It's so confusing.

Yes, because you lack the basic background knowledge to even ask useful questions, let alone to be able to usefully understand the answers. See further comments below.

Before Id spend months or years studying QM. I just need answer to the above questions first so I'd be motivated to study QM textbooks in more details

You've got it backwards. You shouldn't ask questions before you study the textbooks. You need to study the textbooks first in order to be able to ask useful questions at all. If you try to ask questions before studying the textbooks, your questions are going to be vague, unfocused, hard to answer (if they're even answerable at all), and not really helpful to you. Indeed, that seems to describe well the experience you are having here.

 

[jlcd]

His answer, "yes", applies to both questions.

I added the second question after his reply of yes to seek clarifications

Where? Here at PF? Please give a link. I can't usefully comment on the exchange without more context.

See message 16 in https://www.physicsforums.com/threads/hamiltonian-in-qm-for-qft-forces-fields-effects.971043/

What did he mean "This is a very simplified description where scalar quantum fields are replaced by external classical fields. It cannot even describe the motion of a single particle in an external classical electromagnetic field.".

I was describing the quantum Hamiltonian. So was he. Why did he have to mention about replacement of the scalar quantum fields with external classical fields? What was he thinking or context? And why did he state it cannot even describe the motion of a single particle in an external classical electromagnetic field? It got me bewildered for a week and even PSE didn't make it clear so please clarify it before delving so much time on any textbook (in fact i read textbooks many years but forgot most of them because of no specific question or interests). Thank you.

Yes, because you lack the basic background knowledge to even ask useful questions, let alone to be able to usefully understand the answers. See further comments below.
You've got it backwards. You shouldn't ask questions before you study the textbooks. You need to study the textbooks first in order to be able to ask useful questions at all. If you try to ask questions before studying the textbooks, your questions are going to be vague, unfocused, hard to answer (if they're even answerable at all), and not really helpful to you. Indeed, that seems to describe well the experience you are having here.

 

[PeterDonis]

What did he mean "This is a very simplified description where scalar quantum fields are replaced by external classical fields. It cannot even describe the motion of a single particle in an external classical electromagnetic field.".

You were talking about the non-relativistic Schrodinger Equation. You can't model quantum fields using that equation. But you can include potential energy due to external classical fields in the Hamiltonian. However, you can only do this for a very limited set of external classical fields. Technically, those fields include a static Coulomb field and a constant magnetic field, so there are certain types of external classical EM fields that you can use this method for, at least for certain purposes and as long as you are willing to accept inaccuracies due to not being able to model quantum field effects like the Lamb shift. But you can't, for example, use it with a time-varying EM field like an EM wave.

I was describing the quantum Hamiltonian. So was he.

Yes, but "quantum Hamiltonian" just means it is the Hamiltonian for a quantum particle. It does not mean all the stuff that contributes to the Hamiltonian, like external fields, is quantum.

 

[PeterDonis]

It got me bewildered for a week and even PSE didn't make it clear so please clarify it before delving so much time on any textbook

Sorry, you've still got it backwards. You can't expect to properly understand the answers to the questions you are trying to ask, or even ask the right questions, until you have the basic background knowledge you will get from studying some textbooks. Even if some of us here are willing to try and reframe your questions into something we can answer, you're still going to be bewildered by the answers, and by the time we have gotten to the point of getting past your bewilderment and reframing all your questions and going back and forth until you feel like you've gotten an answer that satisfies you, we'll basically have given you the equivalent of a basic course in QM using one of those textbooks. But we're not getting paid for this, and we don't have the time to do that level of remedial teaching. At some point you've just got to realize that if you want answers to your questions, you're going to have to put in the time and effort to build a foundation of understanding. There is no short cut.

 

[PeterDonis]

The OP question has been answered. Thread closed.