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Just to update with the answer: In the case of diamond, there's a two atom basis. In this atom basis, we can think of the orbitals as the bonding (filled) and anti bonding (empty) orbitals, which are the ones to give rise to their respective bands. In the case of hydrogen, there's just one atom...

Just found around that each sigma sp3 bond splits into a bonding and an anti bonding state, the former being full and viceversa. The bonding states correspond to the therefore valence bands and the anti bonding to the conductions band. I've got no problem with that, but it seems I have a...

Homework Statement I am asked to discuss the band structure of diamond. I saw the band structure of diamond has 4 filled valence bands and then 4 conduction bands. Silicon, the same. Homework Equations --- The Attempt at a Solution I'm feeling really silly because I don't understand why it is...

@Charles Link Let me ask it to the point: is ##\left(\frac{\partial u}{\partial u}\right)_T=0## (assuming ##u=cRT##)?? I think that could kind of make sense, though I had never considered it. Maybe knowing that I may avoid future problems (but I can't promise it!).

I know keeping T, v constants forbid me from changing the energy, but still I applied the definition correctly, right? The thing is, I know you are right: "the variation of pressure with respect to energy, keeping v and T constants" makes no sense because it is physically impossible. However, it...

Then how is it that ##\left(\frac{\partial P}{\partial u}\right)_{T,v}## can give a different result depending on how I pick it?

The problem I seem to have is the following. Say I want the variation of P with respect to u keeping the v and T constants. Starting from the ideal gas law: $$\left(\frac{\partial P}{\partial u}\right)_{T,v}=0$$ since T and v re constants. However, if I start from ##P=\frac{u}{cv}##, I get...

Hi all. Suppose I have the ideal gas law $$P=\frac{RT}{v}$$If I'm asked about the partial derivative of P with respect to molar energy ##u##, I may think "derivative of P keeping other quantities (whatever those are) constant", so from the formula above I get $$\frac{\partial P}{\partial...

Say we have a Lagrange function with one multiplier a times a constrain. I minimize and solve the system to find a. I now add another constrain to the same system multiplied by the constant b. Is the value of a the same or can it change?

@hilbert2 Say you have a metal surface. How are the bondings between the metal and the liquid?

I don't understand why materials with low surface energy are hydrophobic and viceversa. All I can find are quick phenomenological explanations that don't quite deal with the physical (microscopic) process going on. Could anyone provide a good microscopic picture of why it is that way? What's...

@Demystifier yes I did, @stevendaryl pointed that out. Thanks everyone for the answers, I understood it.

How come?

Yup, I meant number operator for a fermionic state (number of particles equals either 0 or 1). So ##\exp({\hat{n}})=\sum{\frac{\hat{n}^k}{k!}}=\sum{\frac{\hat{n}}{k!}}=\hat{n}\sum{\frac{1}{k!}}=\hat{n} e ##?

What the title says. Acting on a fermionic state with the number operator to a power is like acting with the fermionic operator itself. Does this allow us to define ## \hat{n}^k=\hat{n} ##? Or is there any picky mathematical reason not to do so?

Since there are no bonds at the other side of the surface, external layers of solids are usually closer to the next layer. This process is called relaxation. (Example in picture a here). However, at a lecture I attended the other day it was mentioned that some surfaces present expansion...

If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?

Hi. I'm taking a look at some lectures by Charles Kane, and he uses this simple model of polyacetylene (1D chain of atoms with alternating bonds which give alternating hopping amplitudes) [view attached image]. There are two types of polyacetylene topologically inequivalent. They both give the...

Yes. That's what I understood, but I was wondering if there was a simpler way, or at least a more polished formalism to state it. For many body systems that could easily get out of hand, right? I don't understand what you mean by this, PeterDonis.

Sorry again for the notation. I'll take some time to learn the PF Latex notation. PeterDonis, you were right: 3 energy states, plus spin degeneration, so 6 total states, and you got them right. And I proposed a combination of 2 states of 2 electrons each. However, I should have said that they...

Hi there. Excuse that I write it all in standard text but I don't know how to write it otherwise. I'll get right to the question and try to see if I understand the theory from the answer. Suppose we have a system of two electrons, one spin up and one spin down in a system with 3 possible...

Thanks, you both. I'll keep the suggestions in mind.

Hi. I'll be doing a master's degree in nanophysics and working on electron transport in arrays of qubits. I don't know anything (or barely) about the second quantization and would like a book which covers it, and on condensed matter overall. So far I've been told about Bruus&Flensberg's...

Yes, actually just thought about that after posting it but forgot to clarify it in this thread. Thanks for stating it, though. So scalar fields are just functions of coordinates that are invariant under the set of transformations you consider?

My first option was to do it abroad as well, maybe the UK or Germany, too. I'm glad yours was great. I'm kind of assuming that coming from Greece, which is one of the PIGS just like Spain... the university was much better at Germany; am I right? I did an Erasmus year in Nottingham University and...

Hi. I'm just graduating and will be doing a Master's next year, but as far as I know (which isn't too far), there are always a lot of options for a PhD out there, specially if you have worked with someone who can recommend you. I'm Spanish, so perhaps it's a bit different somewhere else. Either...

What I meant, @haushofer, is whether a scalar field would be any function on the position (event) coordinates that returns a scalar, so that when you transform the space on which it acts, you also transform the field. Example: f(x,y)=x+y Now y'=y; x'=2x; f'(x',y')=x'/2+y' For clearance, what I...

Oh, I see. So scalar fields are those defined as invariant, not anything that returns a scalar value based on position. Would any function of coordinates (the current density wasn't) be then a scalar field?

In a semiconductor, the Fermi level sits in the bandgap, whereas in a metal, it does inside a band (this is, actually, what drives their behaviour). Even different orbitals (different bands) can be at the Fermi level for metals. This high density of states is what allows the "electron sea".

Hi. This question most probably shows my lack of understanding on the topic: why are scalar fields Lorentz invariant? Imagine a field T(x) [x is a vector; I just don't know how to write it, sorry] that tells us the temperature in each point of a room. We make a rotation in the room and now...

This is the main point I was trying to get across yet couldn't... Thanks.

@Orodruin help me get back on track, please, because I think I'm making a non-needed mess out of this, which seems rather simple. Let's state the facts clear: The typical Lorentz transformations which appear in the image on my first comment transform the coordinates of an event in system S to...

It's definitely not a mistake, @Seba Gonzalez M; he is a terrific and scrupulous theoretical physicist, and he goes on later on the notes using it.

Yes, @Orodruin, I know passive means changing the basis and active means changing the vector in exactly the inverse way, such that numerical coordinates match, but that's not what's happening here as far as I can see. The explanation I understand from the notes gives different numerical coordinates.

I know that going to a coordinate system whose origin moves at v gives the same coordinates as physically boosting everything by -v and different to v. But is this ("physically boosting everything by v") what we mean when we say "active boost"? Seems to me rather strange, for active and passive...

I read that, @Orodruin, but my questions remain.

Hi. First, excuse my English. In my lecture notes on classical electrodynamics, we are introduced to the Lorentz transformations: a system S' moves relative to a system S with positive veloticy v in the x-axis (meassured in S), spatial axis are parallel, origin of times t and t' coincide...

Thank you all for your answers. I reckon it's a rather simple question, but I was just beginning to study tensors and couldn't get my mind around it.

Oh, I think I'm getting it. Thinking about the matrix representation, that's just like stating that there are such matrices that can't be written as the tensor product of two vectors?

May I state clearly that this is not a problem I must solve for class, this was just an example written somewhere which suggested we did it (thus why I didn't provide an attempt at solving it, just asking why it is that way). I still can't see why.

Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?

Yes, I know proper distance in the spatial hypersurface depends on the coordinate system. I wasn't discussing that; I was looking for an understanding of angular distance and luminosity distance as proper distances of spatial hypersurfaces (implicitly, for my coordinate system). So angular...

I may not have been clear. What I intuitively look for when asking for "physical distance" is the proper length between the two events at a fixed time. I know this is not measurable, but is what I can conceive as distance for the spatial hypersurface at a fixed time. So George discusses the...

Thanks for the notes, George, they are helpful. However, if physical distance is not uniquely defined, when we talk about "physical volume" in other context (see text below equation 3.3.80 of Baumann notes, your second link), what are we talking about? Edit: the line element of spatial...

Hi, Etherington't reciprocity theorem states that distances measured by angular separation and by luminosity differ. My question is which one (if any of them) is the actual distance. I can understand they might differ in an expanding universe, but there's still a physical distance in such one...

Hi, On the context of Maxwell's Demon, it's accepted that Landauer's principle (the erasure of a bit of information requires kTln2 of work) gives a solution to the paradox. The erasure of a bit of information can be seen as the modulation of a double well potential, as in the right side of the...

Yes, I know. A process where the memory of the device takes a state (1 or 0 if its a one bit memory) which corresponds to a given physical variable of the system being measured.

I have an answer to the question, but still there's something quite open to me. Erasure means resetting so that it doesn't interfere the device operation. For example, if a device with a one bit memory controls a door and is instructed to "OPEN DOOR whenever a measure gives 1", it wouldn't work...

Dr. Courtney: It's not about whether theories are "really true". They are not, for they are supported by inference and predictions (deductions only work within an inference theory framework). It's about knowing the limits of the knowledge itself. And by "It" I mean the work of the physicist...

Hi. I hope this is the right forum for the question. Landauer's principle says that the erasure of one bit of information (1 or 0) requires kBTln2 of heat production. When he talks about erasure, it means "resetting back to 0 (or 1)" after measuring. He says something along "this is needed as...