Are conduction electrons localized in space?

[JoAuSc]

Let's say we use a very simple model consisting of non-interacting electrons in a 3D infinite square well, perhaps a cube of a single metal crystal. If an electron is in a particular energy state, then its wavefunction is spread across the entire crystal.

However, electrons are said to travel through conductors like particles, implying that their position is at least somewhat localized. This would imply that an electron with a somewhat well-defined position must be in a superposition of energy states whose wave functions cancel out everywhere but near the position of the electron.

So, which is it? I'm betting it's the latter one, and that the interactions between an electron and its environment act more as a measurement of its position than its energy, but I'm not sure and I'd like to hear from the experts.

 

[crazy_photon]

I'm by no means an expert, but i'll put my 2 cents worth.

In my opinion its the former, i.e. electrons are de-localized in space (localized in momentum space). This is the only way that the notion of energy-band picture makes sense to me, along with Fermi-Dirac statistics. The electron 'moves' by changing its wavevector, i.e. in a case of infinite square well the excitation hops between the quantum number n describing the ladder of states.

Lol, I might get myself in trouble right now, but at least it hopefully will generate interesting discussion...

 

[ZapperZ]

Let's say we use a very simple model consisting of non-interacting electrons in a 3D infinite square well, perhaps a cube of a single metal crystal. If an electron is in a particular energy state, then its wavefunction is spread across the entire crystal.

This makes no sense!

The potential you're describing has nothing to do with the potential of a solid, much less, a metal. So why do you think you can use that to represent a metal?

And why are you inventing such a thing for a metal? What's wrong with the periodic potential that gives you the Bloch wavefunction?

Zz.

 

[crazy_photon]

What Zz says is definitely true -- i.e. one has to take into account crystalline lattice. However, one can think of zeroth order model of the solid is that of a potential box, corresponding to the edges of the solid. I thought the question wasn't really about that, but whether the conduction electrons (say they are in this box if that's what we agree to use as a model) are localized in space or not.

I will have to go with 'no', because of the reasons provided in my previous post.

I'm new on this forum, but I think i can 'smell' a good discussion coming...

 

[ZapperZ]

I am not sure there was ever a infinite-potential well that has ever been used to "model" a metal. I can't recall seeing such a thing in any intro QM classes (or did I slept through it?).

Now, if one were to use the plane-wave "free" particle description to model the conduction electrons, THAT I can understand. But then, one then argues "But ZapperZ, what if we make the boundary of the infinite potential infinitely far away?"

Viola! Plane-wave free particle! So why go through the hassle of infinite potential set up?

Zz.

 

[jostpuur]

This makes no sense!

The potential you're describing has nothing to do with the potential of a solid, much less, a metal. So why do you think you can use that to represent a metal?

And why are you inventing such a thing for a metal?

I am not sure there was ever a infinite-potential well that has ever been used to "model" a metal. I can't recall seeing such a thing in any intro QM classes (or did I slept through it?).

The free electron model is quite common thing to see in introductions to solid state physics.

What's wrong with the periodic potential that gives you the Bloch wavefunction?

The Bloch waves are as delocalized as plane waves (or as eigenstates in big box potential), so replacing plane waves (or eigenstates in big box potential) with Bloch waves does not change the original question essentially. I'll ask the original question again, this time in a form demanded by ZapperZ:

"Do the electrons exist as localized wave packets, or as delocalized Bloch waves?"

 

[xepma]

The Bloch waves just form a basis of the (relevant part of the) Hilbert space. If we had a single electron it would be in some superposition of these Bloch waves.

But it's not even correct to talk about a single electron inside a metal. There is one, multiparticle state/wavefunction describing all electrons. This wavefunction is antisymmetric with respect to all the electrons. And we use a basis of Bloch waves to describe the part of this wavefunction which is responsible for the conducting properties.

 

[ZapperZ]

But it's not even correct to talk about a single electron inside a metal. There is one, multiparticle state/wavefunction describing all electrons. This wavefunction is antisymmetric with respect to all the electrons. And we use a basis of Bloch waves to describe the part of this wavefunction which is responsible for the conducting properties.

Actually, we CAN. That's the whole point of Fermi Liquid Theory.

You change a single many-body problem and transpose it to many one-body problem. In doing that, you renormalize the charge carrier into a quasiparticle whereby the many-body interaction has been simplified via a mean-field potential.

Zz.

 

[xepma]

Yes, ofcourse. But that doesn't get rid of the statistics. In the end you're still describing a completely antisymmetrized wavefunction. And not a wavefunction which is "just" a product of single-particle wavefunctions. Although you're right that this step is, in some sense, only taken at the end.

 

[hiyok]

I'd like to add several remarks:
(1) It is surely not proper to model metals using infinite high well potential. Because, in this case, the electron can not be in Bloch states, which is based on translational symmetry. And thus, the electrons can not have any values of averaged velocity than zero. In other words, these electrons cannot move at all !

(2) In a real metal, I believe the conduction electrons are better visualized as wave packets. The reason is the interactons of the electron with its environment, and such interactions are random. Actually, this is exactly the picture adopted in the semi-classical theory that deals with the transportation properties of a metal (see e.g., D. Mermin)

hiyok

 

[sokrates]

But it's not even correct to talk about a single electron inside a metal. There is one, multiparticle state/wavefunction describing all electrons. This wavefunction is antisymmetric with respect to all the electrons. And we use a basis of Bloch waves to describe the part of this wavefunction which is responsible for the conducting properties.

Actually, we CAN. That's the whole point of Fermi Liquid Theory.

You change a single many-body problem and transpose it to many one-body problem. In doing that, you renormalize the charge carrier into a quasiparticle whereby the many-body interaction has been simplified via a mean-field potential.

Zz.

Many wrong things have been stated in this thread but this particular one from Zz is a jewel. Read it over and over again, and try to absorb it EXACTLY the way Zz put it.

Very good post.

 

[sokrates]

The free electron model is quite common thing to see in introductions to solid state physics.

With a very big, important, impossible to miss CAVEAT.

That the free electron model could be used ONLY IF the effective mass approximation is valid and that is true only for small energies around the chemical potential.

 

[sokrates]

I'd like to add several remarks:
(1) It is surely not proper to model metals using infinite high well potential. Because, in this case, the electron can not be in Bloch states, which is based on translational symmetry. And thus, the electrons can not have any values of averaged velocity than zero. In other words, these electrons cannot move at all !

hiyok

Wrong.

You don't definitely need Bloch waves to model conduction band electrons. Some kind of an effective mass approach is most frequently used in practical calculations.

And you can assume WHATEVER boundary condition you like (infinite well, PBC, Open, Dirichlet) AS LONG AS the boundaries are far, far away from the region you are interested in. This ensures that whatever you assume for the boundary condition DOES NOT affect the actual transport properties.

It turns out that the math is simplest when PBC (periodic bounary conditions are assumed) which gives k_x= \frac{2\pi}{L_x} kind of periodicity ( allowing negative k-values). But you could just as well go about assuming an infinite well, and in this case
k_x = \frac{\pi}{L_x} but this time negative k-values yield identical states and you end up with the SAME number of states you'd get if you had assumed PBC initially. This is re-assuring.

So don't get things all mixed up. Boundary conditions are completely independent of how you model the solid inside.

Inside the solid, you could choose a free-electron model (provided that you have a valid effective mass description) or you could do it exactly using the Bloch waves, but these have nothing to do with the kind of boundary condition you assume.I know what bothers people when I say you can have a particle-in-a-box and current flow in that box at the same time, because they are thinking in terms of resonant energies ,and they assume discrete energy levels. But this boils down to one of the most difficult problems in physics:

You have the simplest conductor with one single energy (ignoring spin) and you make two contacts to it. How do you model the current flow through this thing?

The answer only became clear in the last 20 years with the rise of mesoscopic physics and some theoretical acknowledgment of the importance of contacts. And the current viewpoint in the community is that when you make contacts to a BOX the levels are not really resonant energies anymore - but they broaden out in infinite range.

And energy becomes and independent variable of the electrons because you can EXCITE the box from the contacts AT ANY ENERGY.

To get more information on this you can google

Quantum Transport
nanoHUB.org
Meaning of Resistance at the Nanoscale,

etc...

 

[hiyok]

Hi, sokrates,

I understand what you mean. But, actually I was not talking about the boundary conditions. I was either not talking about Bloch waves. What I really intended is that, maybe you can have many options of boundary conditions, but it is inevitable that these electrons are in delocalized states, otherwise there won't be any current and there would not be any disputes over the original post any more.

hiyok

 

[Manchot]

This is a subject that I've often found confusing, as I've never been able to reconcile the semiclassical approximation that people use (e.g., the Boltzmann transport equation) with the quantum mechanical structure of a solid. That is, Bloch waves are obviously extremely delocalized, while the BTE requires that particles be localized in position and momentum space. Is it just a matter of constructing the "right" wavepacket? Is it mostly due to the decoherence that interaction with the environment provides? Or is it some combination of the two?

 

[hiyok]

I believe the decoherence should be the sole cause. If one cools a metal to much low temperatures, the semiclassical picture should eventually break down.

 

[JoAuSc]

Thank you, everyone. I didn't expect to go to physicsforums and find a month-old thread of mine with 15 replies.

This makes no sense!

The potential you're describing has nothing to do with the potential of a solid, much less, a metal. So why do you think you can use that to represent a metal?

And why are you inventing such a thing for a metal? What's wrong with the periodic potential that gives you the Bloch wavefunction?

Zz.

I am not sure there was ever a infinite-potential well that has ever been used to "model" a metal. I can't recall seeing such a thing in any intro QM classes (or did I slept through it?).

I don't doubt that an infinite square well is a bad approximation for a solid, missing behaviors such as band gaps and such, but it seems like a good way to discuss whether electrons in solids are localized and thus in a superposition of basis states, or spread out across the solid. I'm using this model since it's the first one mentioned in my solid state physics textbook (Chapter 6 in Introduction to Solid State Physics by Kittel, described as a "free electron gas".) I'm not using the Bloch waves since the sine waves of an infinite square well are much simpler.

Now, if one were to use the plane-wave "free" particle description to model the conduction electrons, THAT I can understand. But then, one then argues "But ZapperZ, what if we make the boundary of the infinite potential infinitely far away?"

Viola! Plane-wave free particle! So why go through the hassle of infinite potential set up?

Zz.

That's a good point. I can't imagine that the choice of boundary conditions would make that much difference, though, and it seems that a metal cube of a finite size would be more accurately described by something with boundary conditions that aren't infinitely far away.

I'd like to add several remarks:
(1) It is surely not proper to model metals using infinite high well potential. Because, in this case, the electron can not be in Bloch states, which is based on translational symmetry. And thus, the electrons can not have any values of averaged velocity than zero. In other words, these electrons cannot move at all !

That would only be true for eigenstates. For a superposition of, for example, the first and second energy states of the 1D infinite square well, <x> shifts back and forth.

The electron 'moves' by changing its wavevector, i.e. in a case of infinite square well the excitation hops between the quantum number n describing the ladder of states.

That brings something to mind. I didn't think about how localized electrons would collide with, say, a phonon, where you start out with two incident wavevectors and end up with two different ones. A localized electron would necessarily have many wavevector components, so in a sense if it was involved in a collision, then part of its wavefunction would be reflected and the remaining part wouldn't be affected, which seems strange. Also, wavepackets spread out over time, so if an electron was indeed located in a certain region, then it'd either need to have its position measured on a regular basis, or it'd spread out.

This is a subject that I've often found confusing, as I've never been able to reconcile the semiclassical approximation that people use (e.g., the Boltzmann transport equation) with the quantum mechanical structure of a solid. That is, Bloch waves are obviously extremely delocalized, while the BTE requires that particles be localized in position and momentum space. Is it just a matter of constructing the "right" wavepacket? Is it mostly due to the decoherence that interaction with the environment provides? Or is it some combination of the two?

Yeah, exactly my problem. There are large gaps in my knowledge of this sort of thing.

 

[crazy_photon]

That brings something to mind. I didn't think about how localized electrons would collide with, say, a phonon, where you start out with two incident wavevectors and end up with two different ones. A localized electron would necessarily have many wavevector components, so in a sense if it was involved in a collision, then part of its wavefunction would be reflected and the remaining part wouldn't be affected, which seems strange. Also, wavepackets spread out over time, so if an electron was indeed located in a certain region, then it'd either need to have its position measured on a regular basis, or it'd spread out.

Thanks for reading my reply. Again, I'm not an expert, but I don't think many people on here are (not intended as an insult). So, i feel justified sharing 'my view' on your question, without fear of being wrong - afterall we're just having a scientific discussion.

The problem of electron-phonon scatter:

Back to the 'constraints' of the question, i.e. your particle(s)-in-a-box model. First off, as many have pointed out, that is not the correct way of describing the problem, but again, in the 'zeroth' order I don't see why not. Again, for discussion sake...

In such a model you get stationary solutions (i.e. standing waves). I'm talking for low enough temperatures that the modes are indeed orthogonal (If I say zero temperature, we don't have phonons). If a phonon 'comes along' (there's no lattice per se, but nonetheless), it will interact with the electron iff both momentum and energy are conserved (of course). So, if there's overlap in energy between the two (say you're interacting with acoustic phonon), the rest is conservation of momentum. Since standing wave has zero momentum I see that two phonons have to be emitted in the opposite directions (of half energy each). In that process electron would 'hop' up or down the ladder (emission versus absorption of phonon) corresponding to the gained energy.

 

[ZapperZ]

That's a good point. I can't imagine that the choice of boundary conditions would make that much difference, though, and it seems that a metal cube of a finite size would be more accurately described by something with boundary conditions that aren't infinitely far away.

Er... Alpha Centauri can be considered to be infinitely far away when you consider gravitational forces acting on the earth. But yet, to our galaxy, it isn't. If you are the size of Angstroms, the edge of a metal cube that is centimeters long is infinitely far away such that it doesn't matter anymore. That is why in many description of the properties of solids, the boundary condition of the surface or the edge of the material is typically insignificant. You don't see the BCS ground state accounting for the size of the superconductor, do you?

Zz.

 

[JoAuSc]

Er... Alpha Centauri can be considered to be infinitely far away when you consider gravitational forces acting on the earth. But yet, to our galaxy, it isn't. If you are the size of Angstroms, the edge of a metal cube that is centimeters long is infinitely far away such that it doesn't matter anymore. That is why in many description of the properties of solids, the boundary condition of the surface or the edge of the material is typically insignificant.

You're arguing that for bulk solids, it physically doesn't matter whether you choose the infinite square well's sine waves or the free particle models's complex exponentials, but that using the latter is much simpler to deal with. I'll concede that, though only the ISW has a non-zero fermi energy. The Kittel text I referred to above, strangely enough, mentions the infinite square well as the model used but uses the complex exponentials in calculations.

You don't see the BCS ground state accounting for the size of the superconductor, do you?

Zz.

I'm going to have to take your word for it.

 

[conway]

I'm probably going to get kicked out of the discussion group for this, but ZapperZ has an awful lot of nerve ridiculing you for modeling the problem with a potential well and then coming back and niggling over the boundary conditions.

 

[ZapperZ]

I'm probably going to get kicked out of the discussion group for this, but ZapperZ has an awful lot of nerve ridiculing you for modeling the problem with a potential well and then coming back and niggling over the boundary conditions.

Read again. Where did I "ridicule" him? I asked because I am utterly puzzled by the thought of going around the long way to get to something that could easily be reached directly. The boundary condition IS the whole issue here with regards to the situation.

Look at the BCS ground state, the Drude model, the Bloch wavefunction, etc... etc. Many of these (and quite a few of them you use as basic electric circuit laws such as Ohm's Law, etc.) simply do not CARE about such boundary conditions. This is why I queried on the whole rational of considering it in the first place. Is there some insight in imposing such boundary that gives us something that the traditional simple model doesn't do? I haven't seen such explanation yet.

So how is this ridiculing? If you think it is that, you ought to attend a lot more physics seminars, or go attend the APS March Meeting. Compare to those, what I've said here is a non-event!

Zz.

 

[sokrates]

I'm probably going to get kicked out of the discussion group for this, but ZapperZ has an awful lot of nerve ridiculing you for modeling the problem with a potential well and then coming back and niggling over the boundary conditions.

Since when straight intellectual bashing is called "ridiculing"?

If you took the time to read the entire thread, you would have seen that quite a few of us have been perpetuating the same things over and over again.

There's no such thing as modeling the problem with an infinite potential well. Dynamics -- and boundary conditions are two different things. These should not be mixed up.

You cannot just go about choosing a free electron model inside the device, unless you could be working under the conditions of the EFFECTIVE MASS theorem. This would not be an atomistic model and the wavefunction actually would be smoothed out over the lattice. But hey, you could just as well use an atomistic tight-binding model, to get the actual Bloch waves and rapidly changing wavefunctions.

These have NOTHING to do with what happens at the boundaries UNLESS you have a nanostructure where the boundaries are so close that they actually matter.

There's only one place that I know of , where you can sense a difference experimentally:

Graphene Nanoribbons (GNR) and Carbon Nanotubes (CNT):

CNT is the only material in nature that PBC (periodic boundary conditions) really exists -- and in GNR , the boundary conditions are really like that of an infinite well (because the lattice abrubtly ends) and the difference is obvious:

In GNR you see quantized conductance steps of 2 e^2/ h (quantum of conductance including spin) , and in CNT you see quantized conductance steps of 4 e^2 /h instead.

This is due to the familiar pi / a and 2*pi/a spacing of states depending on the boundary conditions I described in detail a few posts earlier.

Now the Hamiltonian of GNR and CNT is almost exactly the same - but since these are really small devices (a few atoms thick) the boundary conditions actually make a difference.

In regard to what ZapperZ is saying: I think this could be the only example where you could see that choosing a different boundary condition changes experimental facts in your device but this is due to the NANO-aspect of the problem.

 

[crazy_photon]

This discussion on applicability of boundary conditions to this problem is amusing to me. Was I the only one that interpreted the original post to mean that this was taken as a "toy" model of a solid, i.e. a 1D box with bounding potential at zero temperature, etc... to boil the problem to the one we can solve...a zeroth order approximation to metal if you will? Sure, this is not in your standard solid-state textbook (i don't remember seeing it in kittel, but i'll take another look...) -- does that mean that it can't be discussed? APS March meetings would be so boring if to all the gedanken experiments somebody was replying - "no, sir, no can do! this is not in my solid-state book!"

Maybe this reality check can help? Consider a 'tiny' cube of metal. Momentum space is quantized along all three dimensions with some separation Dk. Now, you can make your cubes larger if you want, say expand by N in all the directions and sure enough the spacing between the modes will decrease by (1/N) and even go to zero as you go out to infinity. So, let's not be mathematicians and not talk about infinities. That leaves us with the notion that even for your cubes of 1x1x1cm piece of copper the states are separated by the corresponding Dk/N, where N is a large number. The characteristic quantity that allows one to judge the transition from quantized to continuous is the kT. So, if say DE associated with Dk is smaller than kT, then we talk about Drude... So, this corners the discussion into the realm - how large is the structure and at what temperature... errr... the whole point was to talk about basic physics without these hangups... can we get back to the essence of the question of localized versus delocalized, instead of nit-picking about the boundary conditions??

 

[ZapperZ]

This discussion on applicability of boundary conditions to this problem is amusing to me. Was I the only one that interpreted the original post to mean that this was taken as a "toy" model of a solid, i.e. a 1D box with bounding potential at zero temperature, etc... to boil the problem to the one we can solve...a zeroth order approximation to metal if you will? Sure, this is not in your standard solid-state textbook (i don't remember seeing it in kittel, but i'll take another look...) -- does that mean that it can't be discussed? APS March meetings would be so boring if to all the gedanken experiments somebody was replying - "no, sir, no can do! this is not in my solid-state book!"

Then you have completely missed what I was asking.

I was asking for the rational reason on WHY one would want to consider such a thing, when it makes the problem so much more DIFFICULT!

We continue to teach the wave picture of light when we talk about interference and diffraction, when we KNOW for a fact that we can derive ALL of such phenomena via QM alone without invoking such wave picture (ref: Marcella). But yet, WHY do we continue to use the wave picture whenever we deal with interference and diffraction? Isn't this just MORE confusing to the student? No, because handling it via the wave picture makes it EASIER to deal with, and the outcome and results can be obtained in a more transparent manner. The QM derivation of it is so much more tedious.

So there is a pedagogical reason for continuing with the wave picture. What is the pedagogical reason for adopting the line of attack in the OP? This is what I have been asking, and this is what I haven't been given a sufficient answer to. If all we have here is "I have no idea why I would want to pursue such a thing and it just came out of thin air", then I'll shut up.

Zz.

 

[sokrates]

This discussion on applicability of boundary conditions to this problem is amusing to me. Was I the only one that interpreted the original post to mean that this was taken as a "toy" model of a solid, i.e. a 1D box with bounding potential at zero temperature, etc... to boil the problem to the one we can solve...a zeroth order approximation to metal if you will?

A classical delusion of a beginner...

Just because all the zeroth order elementary QM textbooks start with the amusingly simplistic particle in a box problem to show the beginner that the levels will come out to be quantized DOES NOT mean that it is applicable to the HUGELY complicated solid structures where the picture is NOTHING like that "TOY" model.

A toy model of a solid could be a 1D nanowire, or a periodic array of atoms, or something of that sort but NOT the first thing you see in a QM textbook meant for layman.

No; I am sorry but you cannot use your "model" for this problem. Particle in a box is meant for my little brother who is a freshman... If you think you can get away with that, you are deeply mistaken. There are some problems that require at least some sane 5-minute thinking before you make a "valiant" attempt to come up with an extremely "versatile" model nobody has thought before! You can get lost in your textbooks as long as you want, EVEN Kittel in his elementary text will not do that.

I guess your "sanity check" will be amusing to the working physicist in this field.

Yes, you seem to be the only one to underestimate this extremely difficult problem with a so-called toy model that almost insults a BRANCH of physics, which by the way is the name of the forum under which you are posting, evolving for the past hundred years.

 

[Bob S]

Suppose you had a 1-mm diameter wire, 1 meter long (mass 7 grams), that you used to connect to an ideal battery, and charge it at one amp for 96,000 seconds (Avagadro's number of electrons). You now disconnect the battery and put it in your pocket. Where did all the electrons come from (Did they come from the wire, or somewhere else)? (You charged it with about 25% of ALL the electrons in the wire). What happened to the electrons' wave functions when you put the battery in your pocket?

 

[conway]

No; I am sorry but you cannot use your "model" for this problem. Particle in a box is meant for my little brother who is a freshman... If you think you can get away with that, you are deeply mistaken. There are some problems that require at least some sane 5-minute thinking before you make a "valiant" attempt to come up with an extremely "versatile" model nobody has thought before...

Actually, I thought it was quite a good model to choose for the problem posed by the OP: namely, whether electrons in a metal are localized. The simple model of a potential well would seem to potentially offer some useful insights on this question. I haven't seen any of the naysayers offer any explanation as to why it might be inadequate.

 

[ZapperZ]

Actually, I thought it was quite a good model to choose for the problem posed by the OP: namely, whether electrons in a metal are localized. The simple model of a potential well would seem to potentially offer some useful insights on this question. I haven't seen any of the naysayers offer any explanation as to why it might be inadequate.

Er... derive Ohm's law out of it, for example!

Furthermore, how is it possible to know that a model that doesn't offer any result that matches observation, is somehow a valid model to a particular system? How are you to know that an arbitrary model that was picked is a valid model to offer a "useful insight" on the locality of electrons in a metal?

Zz.

 

[crazy_photon]

The more i think about it the more i come to realization that it's in fact the most correct (from theoretical standpoint) model to describe the problem, of course with the caviat that the lattice is missing.

So, if we turn on the lattice we have the translational periodicity that would be reflected in the wavefunctions, modifying them to Bloch states. Then one sees that dispersion relation for small values of quasi-momentum (k) stays parabolic-like (allowing you to treat the electrons as nearly-free (with renormalized mass)). For large enough k's you run into Bragg scattering with the lattice periodicity, -- the notion of a bandgap. So where are the boundary conditions of the solid come in, i.e. that infinite-square well potential, our 'zeroth' order model? Well, it comes in the fact that the dispersion relation is not continuous, but consists of discrete values of k (much much smaller than pi/a, say epsilon), where a is the lattice periodicity). These discrete values are nontheless there, where you want it or not. Of course at finite temperature (when kT > epsilon^2/2m), this discretization is washed out by the thermal fluctuations.
So, if we either lower the temperature enough for kT to be comparable to epsilon^2/2m, or if we reduce the size of the box, we will notice the effect of boundary conditions on the problem. In what sense? In the sense that we no longer have Drude-like picture, but electrons have discrete energy values, so putting restrictions on what scattering processes can occur and hence modifying all the macroscopic observables - conductivity for example. Another way of saying this is that we will form sub-bands within say valence bands, i.e. mini-bandgaps.

So, what I'm saying is that if you insist to treat the problem from jellium model perspective, go ahead, but if you want to recover proper physics either at smaller dimensions or lower temperature than boundary conditions are very important.

I still haven't seen any opponents offer any answers to the original question of the post: localized versus delocalized states.

BTW, i might be totally dillusional about this due to the fact that I'm a beginner, but isn't that the point of this forum is to share your expertise with others so they can learn from you instead of petting your own ego by putting everybody down? If i want to deal with ego problems i can go talk to professors... or wait... maybe you're one of them?

 

[crazy_photon]

Suppose you had a 1-mm diameter wire, 1 meter long (mass 7 grams), that you used to connect to an ideal battery, and charge it at one amp for 96,000 seconds (Avagadro's number of electrons). You now disconnect the battery and put it in your pocket. Where did all the electrons come from (Did they come from the wire, or somewhere else)? (You charged it with about 25% of ALL the electrons in the wire). What happened to the electrons' wave functions when you put the battery in your pocket?

I think all the wavefunctions will spill out causing a big entangled mess in your pocket

 

[sokrates]

Actually, I thought it was quite a good model to choose for the problem posed by the OP: namely, whether electrons in a metal are localized. The simple model of a potential well would seem to potentially offer some useful insights on this question. I haven't seen any of the naysayers offer any explanation as to why it might be inadequate.

Model? What model are you referring to apart from the boundary conditions? The model is so inadequate, the arguments are so immature that I don't know where to start.

Have you ever seen an E-k diagram of a solid?

 

[conway]

So how is this ridiculing? If you think it is that, you ought to attend a lot more physics seminars, or go attend the APS March Meeting. Compare to those, what I've said here is a non-event!

Zz.

Yes, that nasty, mean-spirited attitude you refer to is certainly part of the culture of professional physicists. I've encountered it before and I don't understand the reason for it. It's certainly nothing to be proud of.

 

[sokrates]

BTW, i might be totally dillusional about this due to the fact that I'm a beginner, but isn't that the point of this forum is to share your expertise with others so they can learn from you instead of petting your own ego by putting everybody down? If i want to deal with ego problems i can go talk to professors... or wait... maybe you're one of them?

I am not a professor. But my job is to understand the details you are glossing over during my Ph.D. Maybe you are doing your own Ph.D. But why don't you propose toy models and provide insights in a field where "you" are the expert? Otherwise, your fame won't get you too far away from physicsforums.

I am sorry but I am not reading your technical explanations ( I just don't have enough time), you need to start from a Solid State Book to even start asking relevant questions instead of "proposing" models, especially at this stage.

 

[sokrates]

I still haven't seen any opponents offer any answers to the original question of the post: localized versus delocalized states.

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

 

[crazy_photon]

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

Well, thank you for sharing all of your insights. Perhaps you can suggest me some books to start from? BTW, i figured you're doing research on nanotubes/nanowires from your earlier posts - can you share your insights as to how the boundary conditions change the problem there?

Also, when you referred to 'E-k diagram', in physics that's commonly referred to as dispersion relation, just thought I'll point it out in case you'd like to read my post with 'technical explanation' at your leisure.

 

[ZapperZ]

Yes, that nasty, mean-spirited attitude you refer to is certainly part of the culture of professional physicists. I've encountered it before and I don't understand the reason for it. It's certainly nothing to be proud of.

Or maybe you misinterpret someone questioning the rational for your actions as being "ridiculing".

When your funding request has been denied because your feelings were hurt when you are asked to show the reasons why something "... maybe interesting, but is it IMPORTANT?", then come back and tell me that this is due to a ridiculing.

And note that you didn't even address my rebuttal on why I considered my questions on the need for such a model as NOT out of line here. I spent time and effort explaining myself, and all I got was this nasty one-line attack. Have you ever considered that maybe YOU are doing the very same thing that you are criticizing?Zz.

 

[conway]

Or maybe you misinterpret someone questioning the rational for your actions as being "ridiculing".


Zz.

Perhaps. I think I can usually tell, but you never know.

 

[Manchot]

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

I don't think it's a trivial question at all. As I mentioned before, while it is certainly the case that Bloch states are delocalized, the semiclassical Boltzmann transport equation can also be used to model electron densities.

 

[jensa]

Let me try to rephrase, what I interpret part of the question to be, in a different way. We know that for very small samples we enter the mesoscopic regime where, due to boundary effects, energy levels are quantized but more importantly the energy eigenstates are (under ideal conditions) well described by standing waves. This is precisely the situation that has been asked about in this thread. Now we "know" that when the size of the sample increases boundary effects become no longer important and we may consider only the bulk part of the Hamiltonian, the eigenstates of which (provided a single particle picture is still relevant) are plane waves, Bloch waves or other more complicated states (depending on how difficult you want to make the problem). The question now is WHY do the boundaries play a lesser role when the size of the system increases? And what determines the characteristic length scale at which the boundaries become unimportant?

Obviously when the size increases the level splitting becomes smaller eventually leading to a continuum of states. This, however does not explain why the bound states can be effectively replaced by plane waves (or bloch waves). In fact it seems quite counter-intuitive since these states are highly delocalized and thus should "feel" the boundaries no matter how far away they are. As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified. Of course I could be wrong, the fact that the experts did not provide this explanation makes me wonder.@Manchot

You may, or may not, know this but I can say that the dynamical equation which governs the wigner function (wigner transform of the density matrix) reduces to the Boltzmann equation in the classical limit. More generally the dynamical equation which governs the wigner transform of the keldysh part of the single particle GF is, in the quasiclassical limit, known as the quantum Boltzmann equation and also reduces to the classical Boltzmann equation in the classical limit (The scattering integral is obtained from the self energy which of course depends on the particular scattering processes involved). The basis states used in this derivation are plane waves. While this is a technical comment, if you want to find out the relationship between delocalized states and the Boltzmann equation, this is probably where you should look.

 

[ZapperZ]

I don't think it's a trivial question at all. As I mentioned before, while it is certainly the case that Bloch states are delocalized, the semiclassical Boltzmann transport equation can also be used to model electron densities.

Actually, unless I'm forgetting my solid state physics classes, the Boltzmann transport equation is purely classical, very much like the Drude model. They both consider free electron gas acting as classical ideal gas. So the comparison here with Bloch wavefunction isn't entirely kosher.

Zz.

 

[JoAuSc]

As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified. Of course I could be wrong, the fact that the experts did not provide this explanation makes me wonder.

Thank you, I think you've provided the best answer so far. I'll look into what you said to Manchot about the relationship between quantum mechanics and the Boltzmann equation.

Actually, I thought it was quite a good model to choose for the problem posed by the OP: namely, whether electrons in a metal are localized. The simple model of a potential well would seem to potentially offer some useful insights on this question. I haven't seen any of the naysayers offer any explanation as to why it might be inadequate.

Ditto.

Perhaps it would be clearer if I asked a couple of related questions:

1. How can we reconcile the image of electrons as particles bouncing around like billiard balls with a certain average collision time with the image of wavefunctions which are spread across the solid, if the wavefunctions are indeed that delocalized?
2. Are conduction electrons typically in one of the basis states or in a superposition? Assume that the electron-electron interaction is negligible, or if that's a looney assumption please let me know why.

Now my query has no reference to any model whatsoever, and people can stop complaining that it's too simplistic.

(For the record, I checked my Kittel text (Introduction to Solid State Physics, 8th ed.). Chapter 6, on the free electron fermi gas, starts on pg. 133. On pg. 134, he introduces the 1D infinite square well, bounded at 0 and L, and lists the eigenstates and energies, including the fermi energy. On pg. 137, when he starts with three dimensions, he lists the sine wave solutions to the 3D ISW but then says that it's convenient to use periodic boundary conditions and thereafter uses plane wave states. I mention this because many here doubt that the infinite square well could be used to model a solid, for example:

There's no such thing as modeling the problem with an infinite potential well. Dynamics -- and boundary conditions are two different things. These should not be mixed up.

I'm not doubting that the periodic boundary conditions make things pedagogically simpler, as ZapperZ said. Btw, I should note here that in my previous post, I said that only the ISW had a non-zero fermi energy. I was assuming that the plane wave model had boundaries infinitely far away and had a finite number of particles, which in retrospect was a silly assumption.)

A classical delusion of a beginner...

Just because all the zeroth order elementary QM textbooks start with the amusingly simplistic particle in a box problem to show the beginner that the levels will come out to be quantized DOES NOT mean that it is applicable to the HUGELY complicated solid structures where the picture is NOTHING like that "TOY" model.

Delocalized, almost a trivial question. Don't worry, it's not an unsolved mystery among us. Just read a few pages of a solid state text, and you'll figure.

So... wait... my "simplistic" model can't be used for my "trivial" question?

Even if what you say is true, you're being a jerk. You insult me and the person you're replying to numerous times. You assume I'm making "a 'valiant' attempt to come up with an extremely 'versatile' model nobody has thought before", rather than using the simplest model mentioned in my textbook to illustrate a quite general question. You say the model is suitable for the beginning of a QM book for layman (I didn't know laymen knew what the infinite square well was). You say it's at the level of freshmen. You say "if you think you can get away with that, you are deeply mistaken". You assume that what I'm doing is almost an insult to solid state physics, as if one could insult a subject by misunderstanding it.

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

 

[ZapperZ]

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

Using your model:

1. Derive Ohm's Law
2. Derive the temperature dependent resistivity of conductors

Shall I go on?

I don't quite get this fascination with "localization" or "non-localization". This is the only criteria you are going by in which a model for a conductor would be considered to be valid? How about being able to match some observed behavior? When will that come in? Next week?

You will note that I had already asked about this already. We are talking about standard conductors here, in which the properties are VERY well-known. ANY proclaimed model to be considered must (i) show some resemblance to a few observed properties AND (ii) claim superior pedagogical simplicity over the more extensive description (see example I gave about the wave picture).

I am still not quite sure why are barking up this tree.

Zz.

 

[JoAuSc]

I don't quite get this fascination with "localization" or "non-localization". This is the only criteria you are going by in which a model for a conductor would be considered to be valid?

Forget the models. I mentioned the infinite square well model as an example of a model where electrons are delocalized, but that doesn't mean I'm wedded to it. I used it to show people my basis for believing that electrons were spread across the solid. For that purpose, I assumed the ISW model would show that aspect as well as any other, but without the extra details needed for this model to be valid in other ways. Maybe I was wrong to assume those extra details would have nothing to do with whether an electron's wavefunction was spread across the solid or just in a small region, but I don't know. I could've used the plane wave model, but it didn't come to mind.

That having been said, let me restate my question: "Are conduction electrons localized in space?" Feel free to answer based on what you know about real metals or realistic models.

 

[ZapperZ]

Forget the models. I mentioned the infinite square well model as an example of a model where electrons are delocalized, but that doesn't mean I'm wedded to it. I used it to show people my basis for believing that electrons were spread across the solid. For that purpose, I assumed the ISW model would show that aspect as well as any other, but without the extra details needed for this model to be valid in other ways. Maybe I was wrong to assume those extra details would have nothing to do with whether an electron's wavefunction was spread across the solid or just in a small region, but I don't know. I could've used the plane wave model, but it didn't come to mind.

But see, this what I really, really do not understand. If your intention was to show that electrons in metals are delocalized, then what is the problem with looking at Chapter 1 of Ashcroft and Mermin, adopt the free electron plane wave model, and go home? Aren't the simplistic plane-wave solution already show that the electrons are delocalized? It is so easy and so obvious because this is QM 101. Why do we bother with the infinite square well model?

Do you now see why I was utterly puzzled with this? You are going from A to B, not directly, but rather in a rather circuitous manner that I find rather unnecessary. I've tried several times to understand the rational behind wanting to do it this way, I haven't seen any.

That having been said, let me restate my question: "Are conduction electrons localized in space?" Feel free to answer based on what you know about real metals or realistic models.

They are not. Based simply on the bloch wavefunction, one can already see that it can't.

Zz.

 

[JoAuSc]

But see, this what I really, really do not understand. If your intention was to show that electrons in metals are delocalized, then what is the problem with looking at Chapter 1 of Ashcroft and Mermin, adopt the free electron plane wave model, and go home? Aren't the simplistic plane-wave solution already show that the electrons are delocalized? It is so easy and so obvious because this is QM 101. Why do we bother with the infinite square well model?

Do you now see why I was utterly puzzled with this? You are going from A to B, not directly, but rather in a rather circuitous manner that I find rather unnecessary. I've tried several times to understand the rational behind wanting to do it this way, I haven't seen any.

I see why you were puzzled.

They are not. Based simply on the bloch wavefunction, one can already see that it can't.

Zz.

Could you elaborate? Does this assume that each electron exists in a definite energy rather than a superposition of different Bloch waves?

 

[ZapperZ]

Could you elaborate? Does this assume that each electron exists in a definite energy rather than a superposition of different Bloch waves?

Er.. we were talking about nonlocalization, no?

All you need to do is to see if you can find <r>, i.e. the expectation value of the position. As with plane-wave function, you'll end up with the same situation.

Zz.

 

[crazy_photon]

As far as I can tell the answer to the question, which has been alluded to in some posts, is that the electron has a finite coherence length, which is the relevant length scale which should be compared to the system size. When the coherence length of the electron is smaller than the system size the electron is no longer able to interfere with itself to form bound states, and a plane wave approximation becomes justified.

This is the same explanation that i gave in my posts, i.e. kT (thermal quanta) in comparison with energy separation.

I can elaborate:

for a given size there's energy separation between the modes allowed by the boundary conditions (if size of the solid is very small, then we have to talk about lattice periodicity instead). So, there's a 'grid' of allowed states defined by the boundary conditions (ISW). If kT overlaps with N of those states, then 'electron' occupies those states and hence (by Fourier transform) starts being localized (in real space) -- this is the high-temperature limit of Boltzmann, Drude, etc... (billiard-ball models). However as we keep lowering the temperature kT decreases until only few or even one of the k states overaps with it - then we are in highly delocalized regime. Another way of saying the same thing: at high temperature electrons lose coherence very fast, however as temperature is lowered, coherent effects start to dominate, i.e. if electron is in well defined k-state its delocalized in real space.

From this also follows that electrons are *not* delocalized in Drude model (as some suggest here). In classical approximations they are treated as billiard balls (localized in space). The confusion with plane wave might be from the fact that its just the basis in which localized state can be represented.

 

[sokrates]

Even if what you say is true, you're being a jerk. You insult me and the person you're replying to numerous times. You assume I'm making "a 'valiant' attempt to come up with an extremely 'versatile' model nobody has thought before", rather than using the simplest model mentioned in my textbook to illustrate a quite general question. You say the model is suitable for the beginning of a QM book for layman (I didn't know laymen knew what the infinite square well was). You say it's at the level of freshmen. You say "if you think you can get away with that, you are deeply mistaken". You assume that what I'm doing is almost an insult to solid state physics, as if one could insult a subject by misunderstanding it.

Whatever happened to, "Look, you can't answer your question with that model, and here's why..."?

Being insulted is something that is "perceived". I certainly did not attempt to give that impression so I am not going to apologize for being harsh. Oh, by the way, I am not insulted by you calling me a JERK - that's usually how INTELLECTUAL discussions end for the losing side.

Finally, I am very happy that you took some time by YOURSELF, going through Kittel or Wikipedia to sustain your arguments and to fill the gaps in your knowledge instead of perfunctorily stating what you know about a very ADVANCED concept and waiting to be cherished for that. I am sorry but I am not your high school teacher. This is the whole point. Do a little homework before coming up with brilliant ideas. No, I don't want to be "extra" kind because your attitude is wrong. You start by PROPOSITIONS and MODELS instead of very mild suggestions or questions on a topic that you obviously did not spend as much time.

You are mixing up two issues once again, by cutting and pasting my irrelevant posts. Let me clarify what I said, if you care to read it carefully this time:

Electrons in a conduction band are OBVIOUSLY delocalized, that is the whole point of free-electron and almost-free electron models. That part is obvious. [[ Did you honestly know what localization meant before ZapperZ's posts, by the way, I am just asking?]] HOWEVER, the concept of current flow and QUANTUM MECHANICAL models that get you to OHM'S LAW are NOT obvious. Not trivial. Even Ohm's law breaks down when L goes to zero. How does it break down ? What happens at the nanoscale? Have you ever thought of taking L to zero in Ohm's Law? Does the conductor become resistanceless? So if it does, what is the difference between ballistic conductors and superconductors? These questions require a few courses in theoretical physics departments. Can you capture that simply with an infinite square well, or some back-of-the-envelope sanity checks? If you can, please let me know.

If you can't and if what you are GENUINELY interested in is to LEARN, start by saying something like " This is something I DON'T KNOW, can I use a zeroth order such and such model to understand this?" instead of being forceful and clinging to arguments that don't hold. If you think I didn't answer your question in detail, why don't you spend time on clarifying PHYSICS rather than calling people names?

If you do that, I promise you everbody in this forum (including the JERKS) will say
"Look, you can't answer your question with that model, and here's why..."

 

[crazy_photon]

I've been seeing your style of responses and let me try to emulate you a little (just for fun, shall we?).

I think you're the one that needs to go back (waaaaay back) and retake the beginning solid state class where things like Ohms Law (direct consequence of Drude model) stems from the classical description of charges colliding as billiard balls - hence highly localized in space. Even my 9th grader sister knows that! You don't have to believe her though, just take some solid-state text that you might have and read it again... perhaps its been too long?

As for shrinking the dimensions to zero - that's when quantum mechanics has to come in (forget your V=IR!) and all things including boundary conditions and lattice periodicity start quantizing available states in k-space leading to for example discontinuous jumps in macroscopic observables.

So, as I have asked once, I'll repeat myself - from your posts you sound like an expert on condensed-matter physics on the nanoscale. So, can you please enlighten us (instead of asking questions back) as to what DOES happen to say resistivity on the nanoscale and from what physical principles does it follow from? I'd like to learn from the expert instead of being called names... again...