Pauli Principle and Electrons in a Metal

[swooshfactory]

Why must we apply the Pauli Principle to electrons in a metal? Do they share a many-body wavefunction?

The saying is that no two electrons may occupy the same state, but am I allowed to say "well, this electron is at the top of the metal, and there is another one at the bottom, those two are far away and don't need to be anti-symmetrized and can each have the same momentum, k."

Many textbooks just say things like "and we apply the Pauli Principle," but I think I'm missing some piece of the puzzle.

 

[Naty1]

What is it that you think makes metals unique'...is that the basis of your question?




This discussion should answer your question from several different perspectives:

http://en.wikipedia.org/wiki/Pauli_exclusion_principle

[The] stability of large systems with many electrons ... requires the Pauli exclusion principle.[4]

Nothing exempts metals from the Pauli exclusion principle.

 

[swooshfactory]

No, I only meant to name metals explicitly since I know that the electrons adhere to the PEP over the macroscopic distance of a piece of metal.

My question is: a metal is so big, so why can't I say "yes, these two electrons have the same momentum, but are separated by macroscopic distances, so they don't need to be anti-symmetried." do the electrons in a metal share a wave function in a non-negligible way?

 

[tejas777]

No, I only meant to name metals explicitly since I know that the electrons adhere to the PEP over the macroscopic distance of a piece of metal.

My question is: a metal is so big, so why can't I say "yes, these two electrons have the same momentum, but are separated by macroscopic distances, so they don't need to be anti-symmetried." do the electrons in a metal share a wave function in a non-negligible way?

If you have a perfectly crystalline metal, which is below the Fermi temperature, and the electrons in the system form a Fermi liquid, then you can write down many particle wavefunction as the antisymmetrized product of all the single particle states, which can be compactly written as the Slater determinant. It does not matter how far two electrons are separated. I cannot comment what will happen in the presence of disorder.

 

[DrDu]

The point is that although a metal may be very large, the number of atoms increases also proportional to the volume. Hence there are always other electrons sufficiently near to a given one to make Pauli principle relevant.

 

[swooshfactory]

If you have a perfectly crystalline metal, which is below the Fermi temperature, and the electrons in the system form a Fermi liquid, then you can write down many particle wavefunction as the antisymmetrized product of all the single particle states, which can be compactly written as the Slater determinant. It does not matter how far two electrons are separated. I cannot comment what will happen in the presence of disorder.

This is what I was expecting to hear. That each electron influences all of the other electrons.

I'm not sure I believe the post by Dr. Du:
The point is that although a metal may be very large, the number of atoms increases also proportional to the volume. Hence there are always other electrons sufficiently near to a given one to make Pauli principle relevant.

This seems to say that (especially the second sentence) proximity is important regarding the extent to which anti-symmetrization is necessary. I have doubts about this because then perhaps I can add electrons with identical momentum to opposite ends of a metal, if the metal is sufficiently large. The post by tejas seems to reveal what many books gloss over:

you can write down many particle wavefunction as the antisymmetrized product of all the single particle states

In this case, proximity plays no role and if one electron has a given momentum, another cannot be added, no matter the macroscopic distance.


Please, if someone can provide a reference to tejas' statement, I would be grateful. I checked the Fermi liquid theory wikipedia page; it seems this is support, albeit slightly indirect: "Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. "

 

[tejas777]

Please, if someone can provide a reference to tejas' statement, I would be grateful. I checked the Fermi liquid theory wikipedia page; it seems this is support, albeit slightly indirect: "Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. "

The reason I mentioned “Fermi liquid” was because when you are writing down the many particle wavefunction as a product of the wavefunctions of the single particle states you are, in fact, writing down the product of the wavefunctions of the “quasiparticles.” Pauli’s exclusion obviously also applies to metallic systems which are not non-Fermi liquids. I was merely stating my assumptions before jumping into the Slater determinant explanation.

You may have already read this in the Wikipedia article, but here it is again to avoid any gaps in communication: The electrons in a metal (for this example) interact with each other through the Coulomb interaction as well as Pauli’s exclusion principle. In the presence of electron-electron interactions you cannot write the Hamiltonian as the sum of single particle Hamiltonians. Consequently, you cannot write down the many particle wave function in terms of products of the wave functions of single particle states; in other words, we cannot use the commonly used separation of variables in PDEs. Ignoring electron-electron interactions, however, we obtain a set of N decoupled single particle Schrödinger equation PDEs, where N is the number of electrons. Each can be solved separately, and then we take the product of their respective solutions. And then we manually implement the Pauli's exclusion principle by antisymmetrizing this product.

In real systems, where electron-electron interactions do exist, it turns out that there exists an adiabatic transformation from this coupled system to a decoupled system. This was Landau's great insight. A system in which this transformation is valid is called a Fermi liquid, and after the transformation, the single particle states obtained are called quasiparticles. It is because of this correspondence between single and multiparticle states exists, we can use the prescription I described in the previous paragraph in describing a real system. The Fermi liquid theory hides the underlying ugly field theoretic treatments of this many-body problem and let's us get away the single particle treatment.

A very good book covering the Fermi liquid theory is:

https://www.amazon.com/dp/0201328240/?tag=pfamazon01-20

A description which is very physically intuitive, without much math, can be found in the first few pages of:

https://www.amazon.com/dp/0486670473/?tag=pfamazon01-20

This is probably too complicated if you’re new to Fermi liquid theory, but here is the classic reference covering field theoretic methods of the Fermi liquid theory (for future reference):

https://www.amazon.com/dp/0486632288/?tag=pfamazon01-20

Oh, I almost forgot to mention: the reference to the statement in my earlier post, which I elaborated upon in the first paragraph, can also be found in, maybe not in a lot of detail (Marder is super compact), sections 6.1 and 6.2 of:

https://www.amazon.com/dp/0471177792/?tag=pfamazon01-20

 

[DrDu]

This seems to say that (especially the second sentence) proximity is important regarding the extent to which anti-symmetrization is necessary. I have doubts about this because then perhaps I can add electrons with identical momentum to opposite ends of a metal, if the metal is sufficiently large. The post by tejas seems to reveal what many books gloss over:

you can write down many particle wavefunction as the antisymmetrized product of all the single particle states

In this case, proximity plays no role and if one electron has a given momentum, another cannot be added, no matter the macroscopic distance.

The point is that momentum eigenfunctions (Bloch functions) are completely delocalized so that it does not make sense to say that two such functions are at opposite ends of a metal.
However a wavefunction for electrons in a metal cannot only be built up an anti-symmetrized product of Bloch functions but also, alternatively, as an anti-symmetrized product of localized functions (Wannier functions: http://en.wikipedia.org/wiki/Wannier_function). The overlapp of the latter determines whether the anti-symmetrization will produce physical effects.

 

[swooshfactory]

I have more information now, for sure, but I still don't understand.

a) Does proximity matter for electrons in a metal (I stick to the example of electrons in a metal because I am familiar with it, and then I can try extrapolating to other systems)?
tejas seems to say no (apologies if I'm misinterpreting this): Each [decoupled PDE] can be solved separately, and then we take the product of their respective solutions. And then we manually implement the Pauli's exclusion principle by antisymmetrizing this product.
DrDu seems to say that it depends on mathematical framework (physically this shouldn't be true, and I'm sure this isn't what DrDu intends, just how I'm interpreting it): The point is that momentum eigenfunctions (Bloch functions) are completely delocalized so that it does not make sense to say that two such functions are at opposite ends of a metal...The overlapp of the [localized Wannier functions] determines whether the anti-symmetrization will produce physical effects.
Maybe I can hone in on my question more, now:

I understand band structure as resulting from the Pauli Exclusion Principle, as well as from periodicity of the lattice, in saying that out of so many allowed quantum states, in general, lowest energy states should fill first, with no two particles/quasiparticles (perhaps I should just say particles) occupying the same state.

When I say "no two" though, what am I actually saying? Am I saying that for any electron I choose to look at, it must not occupy a state already occupied by another electron "in the metal"? Or am I saying that it must not occupy a state already occupied by another electron in its vicinity?

 

[DrDu]

a) Does proximity matter for electrons in a metal (I stick to the example of electrons in a metal because I am familiar with it, and then I can try extrapolating to other systems)?
tejas seems to say no (apologies if I'm misinterpreting this): Each [decoupled PDE] can be solved separately, and then we take the product of their respective solutions. And then we manually implement the Pauli's exclusion principle by antisymmetrizing this product.
DrDu seems to say that it depends on mathematical framework (physically this shouldn't be true, and I'm sure this isn't what DrDu intends, just how I'm interpreting it): The point is that momentum eigenfunctions (Bloch functions) are completely delocalized so that it does not make sense to say that two such functions are at opposite ends of a metal...The overlapp of the [localized Wannier functions] determines whether the anti-symmetrization will produce physical effects.

The effect of anti-symmetrization can be understood in either picture, as they are equivalent.
I just wanted to point out that due to the Heisenberg uncertainty relation it does not make sense to speak of localized momentum eigenfunctions. If you want to discuss the effects of anti-symmetrization in a local picture, it is easier to consider localized eigenfunctions.

Thinking in terms of momentum eigenfunctions, you get the following picture:

If the density is low, the range of the momentum values of the filled Bloch wavefunctions will be narrow, if it is high the range will be broad. In that sense at low densities the effects of anti-symmetrization are less pronounced than at higher densities. To be more precise in the limit of vanishing density and fixed temperature, the Fermi-Dirac statistics for the filling of the Bloch functions will converge to the ordinary Maxwell Boltzmann distribution which also results for non-antisymmetrized wavefunctions.
While the decoupling of the PDE's is very convenient (and leads to the Bloch wavefunctions), the antisymmetrized product of the one particle wavefunctions is invariant under unitary transformations of the one particle functions, in particular under a transformation which leads to the localized Wannier wavefunctions.

There is another point: Once electronic repulsion is taken into account, the effects of anti-symmetrization do not only manifest in terms of the Fermi-Dirac statistics but also in terms of the exchange interaction. It can be shown that for localized wavefunctions (according to the scheme of Edmiston and Ruedenberg, to be precise), the exchange energy is minimal. Hence deviations from classical behaviour are easiest to detect when using a localized basis.

Maybe the following wiki article is helpful:
http://en.wikipedia.org/wiki/Localized_molecular_orbitals