Non interacting Fermions satisfy the Pauli exclusion principle

[fluidistic]

This question is more a question I'd ask in a chat rather than formally on paper/forum.

If we take the free electron model, the electrons are considered as non interacting. It is essentially a 1 particle problem where the potential is constant through space. The electrons are not perturbed at all by other electrons. However they still satisfy Pauli's exclusion principle. I do not quite understand how it is possible that they satisfy that principle and yet be totally unaware of each other through the potential.

What's going on here exactly? How to resolve the paradox that Pauli's exclusion principle indicates that each electron is "aware" of the others, while the FEM indicates that they are totally unaware of each other?

 

[Vanadium 50]

First, one cannot be too lawyerly about idealizations. A ball rolls down a frictionless plane? Actually, it would slide and not roll. Being too literal about them means you won't get the benefit of using them.

Second, if the fermions were truly non-interacting, how could you even tell they were there?

Third, the Pauli exclusion principle is a consequence of the antisymmetrization of the wavefunction. That can - indeed has to - happen where there is an interaction between the electrons or not. You can consider it like a boundary condition.

 

[DrClaude]

Even within an idealized picture where particles actually do not interact, this:

What's going on here exactly? How to resolve the paradox that Pauli's exclusion principle indicates that each electron is "aware" of the others, while the FEM indicates that they are totally unaware of each other?

is the wrong way to look at it.

Particles to not need to interact with each other for the Pauli principle to be able to be followed. Consider a simpler case of an helium atom with a single electron in the ground state, with its spin up. The atom captures another electron, and we know by the PEP that, should that second electron also end up in the ground state, it will be spin down. But this is not because the second electron sees the first one and adjusts its spin accordingly. It is because the quantum state where the two electrons have the same spin while being in the ground state does not exists. There is no such state in the universe, therefore we won't find the atom in that state.

 

[fluidistic]

So the answer so far is that there need not be an interaction between the particles for them to satisfy Pauli's exclusion principle.

If I understand well, as long as I can describe the system as a single wave function, that must be antisymmetric because of indistinguishable fermions, that's enough to ensure PIP will hold.

Let's consider a funny strange crazy example. We have two electrons described by a single wave function that extends from the Earth to the Sun. We do not make any measurement so it is in a superposition of states over such an enormous region. If we were to think classically, we could think that in average each electron is far enough away from the other so that there's no interaction and that they have well defined positions. We would not expect interaction, and I would personally think that the PIP would not need to be satisfied. However QM tells us that these 2 electrons still need to satisfy PIP (and that they do not have well defined positions unless a measurement of position is performed), regardless of whether there's an interaction between the two electrons. Does this latter sentence sound correct?

 

[DrClaude]

Yes, the Pauli principle is followed at all time. Consider any two electrons in the universe, then a proper wave function for the two particles must be anti-symmetric with respect to particle interchange.

The fact that this is the case is observed in collision experiments. See the explanation given by Feynman in
http://www.feynmanlectures.caltech.edu/III_04.html

 

[DrDu]

Let's consider a funny strange crazy example. We have two electrons described by a single wave function that extends from the Earth to the Sun. We do not make any measurement so it is in a superposition of states over such an enormous region. If we were to think classically, we could think that in average each electron is far enough away from the other so that there's no interaction and that they have well defined positions. We would not expect interaction, and I would personally think that the PIP would not need to be satisfied. However QM tells us that these 2 electrons still need to satisfy PIP (and that they do not have well defined positions unless a measurement of position is performed), regardless of whether there's an interaction between the two electrons. Does this latter sentence sound correct?

I think the point here is that in this situation you can build up the wavefunction of the two electrons from spatially separated one particle functions as a Slater determinant. Now as the single particle wavefunctions have no overlapp at all, all expectation values will be the same as for a simple Hartree product wavefunction.

 

[fluidistic]

I think the point here is that in this situation you can build up the wavefunction of the two electrons from spatially separated one particle functions as a Slater determinant. Now as the single particle wavefunctions have no overlapp at all, all expectation values will be the same as for a simple Hartree product wavefunction.

But aren't you assuming that the two electrons aren't entangled? I.e. that they never "met" in the past, for example.

 

[PeroK]

But aren't you assuming that the two electrons aren't entangled? I.e. that they never "met" in the past, for example.

In my opinion, you are thinking too classically about electrons in QM. One aspect of being identical is that electrons cannot be marked to distinguish them from each other. There's no concept really of electrons having "met" and being able to recognise each other.

If particles behaved as individual particles, then QM would not be what it is and there could be no PEP without interaction. Instead, particles must obey the rules of QM, which limits the states that systems can assume. The particles do not, in a classical sense, interact and prevent each other from occupying the same state.

To take an example: a helium atom is a quantum system. The elecrons are not distinguishable and you cannot say that one electron got in first and forced the other electron to change its spin. Instead, the system as whole has certain constraints. One constraint is the PEP. I don't think there is any way to describe classically how the PEP is enforced. Instead, it is a QM constraint on the state of a two-particle system and determines what measurements you can get. Remember also that the two electrons are not actually spinning in the classical sense. A measurement of spin in one direction does not imply spin about the remaining two axes.

 

[fluidistic]

In my opinion, you are thinking too classically about electrons in QM.

Quite possibly, indeed!

One aspect of being identical is that electrons cannot be marked to distinguish them from each other. There's no concept really of electrons having "met" and being able to recognise each other.

I agree with the 1st sentence. We can, however, ensure that there are N electrons in a particular region. And of course, we cannot tag any electron and see how it evolves in the pack of electrons, for it is indistinguishable from any other.

Ah, I see what you mean now with your 2nd sentence. Hmm, good point indeed. I'll have to think about it, but yeah it doesn't make sense to say that 2 electrons interacted with each other in the past, unless, possibly, the whole system consists of these 2 electrons only? If there are more electrons then yes I can understand your point easily.

If particles behaved as individual particles, then QM would not be what it is and there could be no PEP without interaction. Instead, particles must obey the rules of QM, which limits the states that systems can assume. The particles do not, in a classical sense, interact and prevent each other from occupying the same state.

That's exactly what I was missing when I created this thread. I thank you here again for emphasizing this point.

To take an example: a helium atom is a quantum system. The elecrons are not distinguishable and you cannot say that one electron got in first and forced the other electron to change its spin. Instead, the system as whole has certain constraints. One constraint is the PEP. I don't think there is any way to describe classically how the PEP is enforced. Instead, it is a QM constraint on the state of a two-particle system and determines what measurements you can get. Remember also that the two electrons are not actually spinning in the classical sense. A measurement of spin in one direction does not imply spin about the remaining two axes.

Thanks for reformulating what Feynman wrote about in DrClaude's link posted above.