Help me understand how does Pauli's exclusion principle work

[colegu]

It's the second time I've posted this message, and my english simply sux, so excuse me if I will not be able to explain my misunderstanding quite clear.

Let's imagine a verry long metal cable, which is heated at one head. The quasi-free electrons at that head, as their energy increases, must change their quantum state, but there must be a way for them to *know* what the other electrons quantum states are, in order not to occupy the same state.

So how does a fermion *know* what states are not possible to ocupy?

 

[marlon]

It's the second time I've posted this message, and my english simply sux, so excuse me if I will not be able to explain my misunderstanding quite clear.

Let's imagine a verry long metal cable, which is heated at one head. The quasi-free electrons at that head, as their energy increases, must change their quantum state, but there must be a way for them to *know* what the other electrons quantum states are, in order not to occupy the same state.

So how does a fermion *know* what states are not possible to ocupy?

Two things :

1) an electron does not "know" what to do. The Pauli principle is an inherent property of fermions. You are asking questions like " does a square know it has corners ?". If you look at the theoretical formalism behind fermions, you will clearly see that your approach is useless.

2) You example is not the best one because you are neglecting thermal redistribution effects to reach the point of thermodynamical equilibrium.


marlon

 

[colegu]

Two things :

1) an electron does not "know" what to do. The Pauli principle is an inherent property of fermions. You are asking questions like " does a square know it has corners ?". If you look at the theoretical formalism behind fermions, you will clearly see that your approach is useless.

The principle is inherent to identical fermions, but what is the difference between the quasi-free electrons from the metal cable , and those who are strongly bounded to the nuclei, or the electrons from the materials surounding the metal cable. is it because their wavefunction is different?

so an answer is not to view identical fermions individually but as a whole

2) You example is not the best one because you are neglecting thermal redistribution effects to reach the point of thermodynamical equilibrium.


marlon

Actually I don't see where is the problem with thermodynamical equilibrium, since it has no importance in what I was asking(I'm not trying to measure temperature or something like this, I just wanted to point out that the energy of some electrons _must_ increase in some sort of way, so there quantum state should modify.
Or, is the exclusion principle availible only at TD equilibrium? If this is true, I must admit I didn't know this.

Thanx for your comment.

 

[marlon]

The principle is inherent to identical fermions, but what is the difference between the quasi-free electrons from the metal cable , and those who are strongly bounded to the nuclei, or the electrons from the materials surounding the metal cable. is it because their wavefunction is different?

Such electrons are all fermions and respect the exclusion principle. With respect to that, there is NO difference.

so an answer is not to view identical fermions individually but as a whole

No, but still, in a conductor or whatever solid/ crystal etc etc...The physical properties of that object, eg continuous phonon emission spectra, arise due to the many atoms interacting all together. The individuality of each atom gets lost and new properties arise.

Or, is the exclusion principle availible only at TD equilibrium?

Nono, ofcourse not.

Look, concerning the history of the exclusion principle : In its first (usually chemical useful) formulation, Pauli POSTULATED the fact that 2 electrons could not be in the same quantum state. Later, in the Dirac version of QM (the traditional formulation) is still kept as a postulate.

Even later on Von Neumann used it to invent quantum statistical mechanics.And it was again Pauli who used indirectly [1] to prove his theorem:the spin-statistics theorem (1940) is proven in the context of QED and a generalization to particle physics has been given by Lueders.

[1] "The Connection between spin and statistics."Phys.Rev.,58,p.716-722(1940) found in:"Wolfgang Pauli:<<Collected Scientific Papers>>",edited by R.Kronig and V.F.Weisskopf (1964,Interscience Pulblishers),Volume 2;p911-918:


marlon

 

[colegu]

Such electrons are all fermions and respect the exclusion principle. With respect to that, there is NO difference.

I'm sort of confused right now, so, given an electron in a crystal with a given quantum state, there is no other electron in the same quantum state in the same crystal or everywhere?

I may sound stupid, but to me these postulates, and physical explanations are more important than mathematical formulas, and I want to have things verry clear.

Thank you once again.

 

[marlon]

I'm sort of confused right now, so, given an electron in a crystal with a given quantum state, there is no other electron in the same quantum state in the same crystal or everywhere?

I did not say that. I just said that the exclusion principle applies to electrons in solids or crystals as well.

I may sound stupid, but to me these postulates, and physical explanations are more important than mathematical formulas, and I want to have things verry clear.

Well, do you think Pauli looked at that differently ? Again, the reference i gave you gives a nice overview of this principle. Yes, originally it was a postulate that describes reality correctly, just like the particle wave duality. In the reference, you can also see a mathematical proof for it. Everything in physics needs to have a mathematical proof that respects the rules of the formalism, so i don't get what you mean by "physical explanations are more important than mathematical formulas". Mathematics IS the language of physics. Can you give me an example where you such physical explanations are more important than mathematical formulas ?

marlon

 

[ZapperZ]

It's the second time I've posted this message, and my english simply sux, so excuse me if I will not be able to explain my misunderstanding quite clear.

Let's imagine a verry long metal cable, which is heated at one head. The quasi-free electrons at that head, as their energy increases, must change their quantum state, but there must be a way for them to *know* what the other electrons quantum states are, in order not to occupy the same state.

So how does a fermion *know* what states are not possible to ocupy?

Your question actually isn't that elementary to answer.

First of all, when we solve for the ideal case of a conductor using such free-electron gas, we make the explicit assumption of "plane wave states" and "very large distances" for the boundary conditions. We then obtain the typical fermion distribution of states. Now, under most circumstances, this is OK because typically, the scale of system in question is large enough that, according to the electron, it is boundless and all it cares about are its immediate environment.

However, this is where your example is no longer applicable and has breeched the assumption made. In an ordinary metal, the electrons have no long range coherence. This is because the scattering with the lattice ions and other electrons would destroy such coherence very quickly. So essentially, the Exclusion Principle is only a localized principle, maybe up to an order of 10^3 lattice constants (this is a very rough estimate). Only when you get a superconducting state would there be such long-range coherence for the entire solid. In the latter case, all the electrons are already in a single state that is non-local.

Secondly, even after mentioning the above, you need to remember that the scattering into the high energy states isn't "static". Electrons continue to scatter in and out of those states due to thermal fluctuations. So cannot assume that only one electron will occupy a particular state and that's that. As with the Cooper Pair states in a superconductor, the electrons continue to scatter in and out of the states, like a game of musical chair. As soon as one scatters out, another one will occupy that state. At equilibrium, these scattering process reaches a statistical equilibrium.

Lastly, you need to remember that all the momentum and energy states are continuous for the conduction band. This means that there are infinitesimally high or lower states that are part of the density of states. So it doesn't take that much to find an available state to scatter into.

Zz.

 

[colegu]

I did not say that. I just said that the exclusion principle applies to electrons in solids or crystals as well.

I got it, but I asked what is the difference between the electrons in the metal and those outside the metal.
The difference must be the wavefunction, so the exclusion principle must apply to identical fermions with the same wavefunction?
I don't think I have made myself clear enough. sorry.

Well, do you think Pauli looked at that differently ? Again, the reference i gave you gives a nice overview of this principle. Yes, originally it was a postulate that describes reality correctly, just like the particle wave duality. In the reference, you can also see a mathematical proof for it. Everything in physics needs to have a mathematical proof that respects the rules of the formalism, so i don't get what you mean by "physical explanations are more important than mathematical formulas". Mathematics IS the language of physics. Can you give me an example where you such physical explanations are more important than mathematical formulas ?

marlon

I would love to see a mathematical proof for this, but if it ain't on the web, I don't think I can look upon the reference you gave me.(I don't think i can find anything like that here at the library)
I agree with you that mathematics is the language of physics, but the mathematical formula only describes a physical reality.
Maybe my example is not so good, but I would say that you can't explain mathematically what mass is, whereas you can make physical explanations about it, to clarify the term(mass is the measure of inertia etc..- only words, no formulas)


Thanx again

 

[marlon]

I got it, but I asked what is the difference between the electrons in the metal and those outside the metal.


The difference must be the wavefunction, so the exclusion principle must apply to identical fermions with the same wavefunction?
I don't think I have made myself clear enough. sorry.

ok, the difference is that an electron outside the metal is a free electron while inside the metal, it makes up the metal. "Making up the metal" means that it has lost it's individual properties (of a free electron) and "joined" one big collective ensemble of electrons. This is how energy bands are born.

I would love to see a mathematical proof for this, but if it ain't on the web, I don't think I can look upon the reference you gave me.(I don't think i can find anything like that here at the library)

Ofcourse it's on the web : http://publish.aps.org/

But you need an admission. Download the paper in your university library.

Maybe my example is not so good, but I would say that you can't explain mathematically what mass is, whereas you can make physical explanations about it, to clarify the term(mass is the measure of inertia etc..- only words, no formulas)

Not true, the fact that you say "mass is the measure of inertia" is just the formula expressed in words. The same thing is done in the paper i gave you reference to.

marlon

 

[Gokul43201]

The principle is inherent to identical fermions, but what is the difference between the quasi-free electrons from the metal cable , and those who are strongly bounded to the nuclei, or the electrons from the materials surounding the metal cable.

I'm sort of confused right now, so, given an electron in a crystal with a given quantum state, there is no other electron in the same quantum state in the same crystal or everywhere?

I think this may be the root of the problem! And this too, is quite a valid question.

There are two ways to think about the above question, which I'll paraphrase as "Do only the free electrons in a metal have to occupy different states due to Pauli, or does this apply to core electrons as well?" (correct me, if this is not your query).

It's important to note that the Exclusion Principle, in it's popular form follows from the more general statement about a symmetry property of the total wavefunction of indistinguishable particles. So, one could perhaps argue that core electrons in different atoms are not indistinguishable, since we can attach an index to the atom's position. This however, makes the approximation that these core electrons are completely isolated from each other - that the wavefunction of say, the 1s electron on atom 1 (within this approximation) has zero amplitude at the position of atom 2. This approximation is only valid in the limit of an infinitely high potential barrier between atoms (which is not far from reality). But within this approximation, the core electrons are no longer indistinguishable.

If you want to be more pedantic, you will note that there exists a large but finite potential barrier between atoms, and that in fact, the 1s levels in different atoms are displaced by a very tiny bit (thus breaking the degeneracy ever so slightly), due to the small probability of the 1s electron tunneling from atom 1 to atom 2. So, in this case, the argument is that the Exclusion Principle applies to the "core" electrons just as it applies to the "free" electrons.

Either way, there is no inconsistency.

 

[colegu]

Thank you all, you all helped my clarify some things.
marlon, i cannot retrieve the reference you gave me since it asks for an user and password, which I don't have.

I'm not studying in US, btw.

 

[marlon]

Thank you all, you all helped my clarify some things.
marlon, i cannot retrieve the reference you gave me since it asks for an user and password, which I don't have.

I'm not studying in US, btw.

But your university has access to the site. Ask a professor or the library. It does not matter whether you study in the US or not.

marlon

 

[colegu]

Now I'm in vacation, so in ~10 days I will ask a teacher here and see what he can do.
thanx a lot.

 

[Sojourner01]

Similar question on pauli exclusion; what is the spatial range of this phenomenon? That is, how far apart do (say) electrons have to be in space to be allowed to occupy the same state?

 

[marlon]

Similar question on pauli exclusion; what is the spatial range of this phenomenon? That is, how far apart do (say) electrons have to be in space to be allowed to occupy the same state?

If you are asking how close electrons can come to each other, i would say that is easy to calculate. You know what potential governs their mutual repulsive interaction, no ?

But, the exclusion principle is always valid. It does not depend on distance though it can be local sometimes. In a free electron gass, for example, we acquire the fermion distribution law by using plane waves to describe the electrons. Such plane waves require very large distances for the wavefunction's boundary conditions. We say that electrons have a lrage coherence in this case.

In a metal, the large coherence is broken because electrons undergo scattering with the lattice ions and other electrons. In this case, electrons only "care" about what happens closely around them. This means that electrons only feel the influence of nearby electrons and are no longer "in touch" with distant electrons. The Exclusion Principle is only a localized principle, maybe up to about 10^3 lattice constants.

marlon