What is the physical system that follows the Pauli exclusion principle?

[Tomer]

Hello guys,

here's something that's been bothering me for a while now:
We know the Pauli exclusion principle states that 2 Fermions cannot be in an identical state.
So then we have systems like solids or free electron gases, and we calculate and form Fermi-Surfaces, based on the fact that the electrons are being distributed to different states.
What I don't understand is - what defines a "system" in which this principle holds?
We could have another gas next to our gas, and the electrons would once again take the states from ground state up.
So how does a physical system "know" its a system, such that two electrons in it cannot have the same states? What if I have two "distinct" electron gasses and suddenly i instantaneously mix them? Would then half of the electron take hold of higher energy states in an instant?

I hope the question isn't vague.

Thanks a lot!
Tomer.

 

[Dickfore]

The number of available states per unit volume in k-space is given by:

$$\frac{V}{(2 \pi)^3}$$

where V is the volume of the system. Thus, for systems with twice the physical size, you may accommodate twice as many fermions within a Fermi sphere of the same radius.

 

[Ken G]

I would say the answer is, the "system" is defined by the physicist, and their goals. Formally speaking, in quantum mechanics, no particle ever has its own wavefunction-- all indistinguishable particles respond to the same unified wavefunction. This can have important ramifications, like the Pauli exclusion principle, or interference in an electromagnetic wave. But in lots of situations, we get away with imagining that the "system" is actually a small piece of the whole, maybe even a single particle (like the "particle in a box" kind of quantum solution). This is simply a choice made by the physicist, like the choice to treat the Earth as a sphere when calculating its gravity. When two Fermi gases of indistinguishable particles are well separated, we can treat them as having separate states and separate wavefunctions, but we can't get away with that if they overlap, or are mixed. It's not that the physics of the situation is different, it is that the idealizations the physicist must make to get good results have changed. This is a fact of physics that is not all that well understood-- we do have formal theories about how things work, but we never use these formal theories, we always apply idealizations to them to get anywhere in physics.

 

[Dickfore]

What if I have two "distinct" electron gasses and suddenly i instantaneously mix them? Would then half of the electron take hold of higher energy states in an instant?

How do you keep the "electron gasses" from flying apart?

 

[Tomer]

Thanks a lot for the reply, Ken G.
I am of course familiar with idealizations. However, in this case, it feels fundamentally different.
If I want to treat a "closed electric system" physically, I might assume there are no electrical fields in this system. The fact that there's in practice an electron somewhere creating a field in the scale of 10^-40 doesn't effect our experiment, because we cannot get results more accurate than 10^-20. So we may just as well treat the system as closed.
However, regarding my question, what I see is a "binary" decision.I'll try to explain what bothers me:
If I have two ideally separated electron gasses, I would then expect to see the same at both of them: In each, the electrons would be distributed throughout the states from lowest energy onward. So we'd have two "ground states", each at each gas, and so on.

However, if the gasses are just "slightly" connected (whatever that may mean experimentally), then suddenly the electrons panic - they're not allowed to be at the same state as their friends, and the distribution should suddenly be completely different, reaching out to much higher energy states.
So by "binary" I mean, that there's a certain limit of interaction between these two systems, in which the Pauli principle, as I understand it, implies a completely different distribution of the electrons. It just feels completely non-continuous. What is then this limit?

If you already answered this, I have failed to understand it apparently. :-) I'd appreciate a further attempt.

Tomer.

 

[Tomer]

How do you keep the "electron gasses" from flying apart?

By using violent infinite well potentials a.k.a Boxes?

 

[Dickfore]

By using violent infinite well potentials a.k.a Boxes?

Well, there you go. This is a good approximation of a system. Everything that's inside those boxes. Because the potentials are infinite, there is no leaking of the wavefunctions outside and of the wavefunctions from the outside.

 

[Ken G]

However, regarding my question, what I see is a "binary" decision.I'll try to explain what bothers me:
If I have two ideally separated electron gasses, I would then expect to see the same at both of them: In each, the electrons would be distributed throughout the states from lowest energy onward. So we'd have two "ground states", each at each gas, and so on.

Yes, you would see that-- but that is also just exactly what you would see if you treat the two gases as being comprised of indistinguishable particles that have a joint wave function that spans both gases and does not even say which electrons are in which box. That is what formal quantum mechanics actually requires, but since you get all the same answers to every predicted observation, we just don't bother to do it in that more complicated way. It is an idealization to treat the gases as separate, but one that does not change the answers in any measurable way. It does, however, change the answers in a calculable way, because you could calculate the exchange terms and tunneling probabilities and so forth-- you just would never bother to do so unless the systems were close together!

Thus, if you had written the initial joint wave function "correctly", bringing the systems together presents no new problems, no electron "panic." It is only if you have chosen the initial idealization that you get a problem when the idealization breaks down-- the panic is that of the physicist who has oversimplified his/her problem!

 

[Tomer]

Well, there you go. This is a good approximation of a system. Everything that's inside those boxes. Because the potentials are infinite, there is no leaking of the wavefunctions outside and of the wavefunctions from the outside.

And that's exactly my point: in practice, no potential is indeed "infinite". It's just an approximation.
So a certain leakage would definitely occur.
If it were electrical fields leakage we were talking about, you'd say "alright, so there's a 10^-40 disturbance - that doesn't change a thing".
But the Pauli principle, as far as I understand it, talks about "systems". And if I then have two very weakly "connected" systems, the Pauli principle should then apply, causing extreme differences from my 2 ideal boxes model. In other words, it feels to me, that since no two systems can be purely separated, but only approximately, everything's connected (the famous cliché), which would mean every electron in the universe has to be in a different state - which doesn't seem to be the case, since when someone in the US does an experiment with an electron gas, it doesn't seem to bother him, at which states the electron gasses in Germany are. But then, when does this principle suddenly stop applying?

 

[Dickfore]

And that's exactly my point: in practice, no potential is indeed "infinite". It's just an approximation.
So a certain leakage would definitely occur.
If it were electrical fields leakage we were talking about, you'd say "alright, so there's a 10^-40 disturbance - that doesn't change a thing".
But the Pauli principle, as far as I understand it, talks about "systems". And if I then have two very weakly "connected" systems, the Pauli principle should then apply, causing extreme differences from my 2 ideal boxes model. In other words, it feels to me, that since no two systems can be purely separated, but only approximately, everything's connected (the famous cliché), which would mean every electron in the universe has to be in a different state - which doesn't seem to be the case, since when someone in the US does an experiment with an electron gas, it doesn't seem to bother him, at which states the electron gasses in Germany are. But then, when does this principle suddenly stop applying?

Well, the overlap integrals are exactly like that - exponentially decaying. So, it won't matter.

 

[Tomer]

the panic is that of the physicist who has oversimplified his/her problem!

hehe, indeed, you are correct by saying it's probably not electrons getting panic seizures. ;)

I'll "contemplate" on whatever you and others write/wrote here for a while, and see if it has really gotten clear.

I appreciate the answers a lot!

 

[Ken G]

And if I then have two very weakly "connected" systems, the Pauli principle should then apply, causing extreme differences from my 2 ideal boxes model. In other words, it feels to me, that since no two systems can be purely separated, but only approximately, everything's connected (the famous cliché), which would mean every electron in the universe has to be in a different state - which doesn't seem to be the case, since when someone in the US does an experiment with an electron gas, it doesn't seem to bother him, at which states the electron gasses in Germany are. But then, when does this principle suddenly stop applying?

Ah, now I understand your issue. There is not just one ground state here-- if you have two widely separated systems (say two atoms), and note that "system" is not a claim on reality, it is a choice made by the physicist, then you also have two ground states. That's not by fiat, the way you would write those states is different-- it is separated in space. When we write the ground state for a particle in an infinite well, we pretend the wavefunction is confined to a finite space, but real space is not so confined-- so you can think of two atoms as a double-dipped potential, and place two indistinguishable electrons into that double-dipped potential, with the further requirement that the history of the preparation dictates that one electron is in one dip and one electron is in the other dip (distinguishing the dips but not the electrons), and you'll get all the same (measurable) answers as if you just have two separate problems with one electron each. The latter is just an idealization of the former.

 

[sophiecentaur]

And that's exactly my point: in practice, no potential is indeed "infinite". It's just an approximation.
So a certain leakage would definitely occur.
If it were electrical fields leakage we were talking about, you'd say "alright, so there's a 10^-40 disturbance - that doesn't change a thing".
But the Pauli principle, as far as I understand it, talks about "systems". And if I then have two very weakly "connected" systems, the Pauli principle should then apply, causing extreme differences from my 2 ideal boxes model. In other words, it feels to me, that since no two systems can be purely separated, but only approximately, everything's connected (the famous cliché), which would mean every electron in the universe has to be in a different state - which doesn't seem to be the case, since when someone in the US does an experiment with an electron gas, it doesn't seem to bother him, at which states the electron gasses in Germany are. But then, when does this principle suddenly stop applying?

Does there really have to be a problem here? If you consider an 'isolated' Hydrogen atom then you get a well defined states / Energy levels. Introduce some more H atoms, so that there are N of them and these states will be split into N, very close and possibly indistinguishable states. Put the gas under high pressure and you will / could observe this effect as a splitting of the spectral lines into N individual lines. Introducing another N atoms, elsewhere is merely introducing a further, immeasurable splitting of the lines that have already been split by the closer N atoms. Is it not just a matter of degree and the precision of one's measurement?
The argument that applied to the very first added atom merely extends to all the others - only less and less, as they get further and further away.

 

[cgk]

here's something that's been bothering me for a while now:
We know the Pauli exclusion principle states that 2 Fermions cannot be in an identical state.

The problem is right here. The correct exclusion principle is not "two Fermions cannot be in the same state", but "The many-body wave function of an Fermionic system has to be antisymmetric." It is clear that the second statement does not suffer from any definition uncertaincies with respect to the system or state (every single Fermion which is interacting in any way with any other Fermion has to be included in the total wave function).

The anti-symmetry requirement of the total wave function boils down to "two Fermions cannot be in the same /orbital/" (a one-particle wave function) in mean-field theories like Hartree-Fock/Kohn-Sham/Extended Hueckel/Other tight binding theories. In those theories you *assume* that the total wave function can be approximated as a single Slater determinant, which allows you to construct the orbital picture of wave functions. But those are approximations! In those cases it is always assumed that you treat the interaction between the particles in a very simplified manner. This let's you get away with many interpretation problems, but when there are uncertainties, like the question of which states are actually meant in the current case, you have to go back to the full wave function picture in order to see what is happening.