also -- in a&m, they say that they are going to improve upon the free electron model by including potential terms that incorporate the electrons influencing each other (eq 17.2) then they say that, in an independent electron pictures, the contribution of an electron level to the charge would be what is given in eq 17.5. so isnt this kind of contradictory? are we moving away from free electron or not? these are the assumptions that give rise to the hartree equations
@Relativisticcucumber It is independent particle approximation; they are free as in not having correlations, such that you don't insist upon using coördinates that enforce correlations, but you may have the antisymmetry in Slater determinant, and in DFT you ignore even the Slater determinant. The Coulomb interaction is too strong, though, so we must not be free to the point of ignoring the other electrons.
@Relativisticcucumber That's going overboard. Condensed matter physics is just difficult, no matter who it is that is attacking the problem. Don't attack the messenger who tried to make things nice for you.
@SillyGoose One of the first things that is done when moving from scalar QFT to spinor QFT, is to show that the spinor propagator is a spinor part modification of the scalar propagator, so that most of the machinery is not wasted.
> Asymptotic scattering theory works well for predictions for many experimental scenarios, especially in particle physics, but does not cover all cases of interest. Rel ativistic quantum information is a feld in which fnite time processes that occur in a local laboratory environment are theoretically and experimentally investigated
this shows that the S-matrix theory view of physics is wrong
when people do scattering in lab, they are ultimately measuring some local observables. i think the S-matrix is an approximate description of those local measurements
Give a Crystalstructure. If we have a general shape for the crystal, say FCC. (Eg: NaCl) Do the individual atoms, form also a FCC? ( in the case of NaCl, both Na and Cl if considered separetly, form a FCC-structure). Considered spearetly meaning, that if you ignore the other points in the strucutre and just look at them. Is this true generally?
@Madder Sometimes yes, sometimes no. For NaCl the Na atoms form an FCC lattice and so do the Cl atoms so yes you can pick any atom as your lattice points.
But there may be structures where the primitive lattice contains two or more of the same atom.
In that case you can pick one of the atoms in the primitive cell as the lattice points, but not all the atoms in the primitive cell.
@JohnRennie Would you be so kind in providing me an example? i find it hard to imagine, since if i am picking a Basis for my lattice. The translational symmetry needs to apply to each component of my basis. So i do not understand how it is possible for an Atom to not create the same structure as the whole Crystal.
As far as i understood, if i take multiple atoms, it is basically the process of forming a Basis. So instead of two points, i "join" them together into one. But this point would have the exact symmetry as the lattice. I will draw it.
So my new Basis. Which is both of these atoms, do form the lattice?
@RyderRude I work, roughly speaking, in condensed matter theory. So of course I find this very interesting. But I also enjoy topics outside my field. Overall, I'd say quantum theory and to some extent the underlying mathematics, many body theory, perturbation theory, statistical physics...
but wouldnt the second atoms you are drawing be also present to the right and under or are they only present in one place? that does not define a crystal. In your example, if we extend the atoms floating nearby to nearby atoms, you can just take them as a basis. i do not understand frankly what you try to imply.
Yes, imagine it as the lines AB then BC drawn as two vectors. If the gradient of AC is less than 1 (spacelike) then either AB or BC must also have a gradient < 1.
@Slereah What's long about it? I only know "It's a cover $\{U_i\}$ such that all finite intersections are either empty or diffeomorphic to Cartesian space", which I wouldn't call "long ass" :P
Can someone read these two things iw rote and tell me i fi made a mistake?
" Def: Unit-cell: a region that fills the point space without any overlapping by translation only. It is spanned by the lattice vectors. Theorem: There are exactly 7 of those. They all Parallelepipeds. They are called Crystal Systems. "
I am trying to give precise definitions of these concepts. Sadly most books are very vague about it.
I should probably add also without gaps. But thats implied.
If one considers two states $|\psi_1\rangle$, $|\psi_2\rangle$, which can be expressed as linear combination of the eigenstates of the Hamiltonian, would it be incorrect to say that the expectation value $\langle \psi_1|H|\psi_1\rangle=\langle \psi_2|H|\psi_2\rangle$ ?
The Book defines a primitive unit cell as a parallelpiped spanned by lattice vectors, and then shows this example for "different choices of unit cells". (c) is clearly not spanned by any vectors!! In (a) and (d) The vectors would point at empty space. Which is a contradiction to the definition for a lattice vector...
@imbAF What is the relative clause doing there? All states can be expressed as linear combinations of eigenstates of the Hamiltonian. We've been over this.
@imbAF You can ask yourself the simplest case. is $\langle n_1 \lvert H \lvert n_1 \rangle = \langle n_2 \lvert H \lvert n_2 \rangle$ in general? (nondegenerate eigenstates are themselves trivial linear combinations of eigenstates).
@SillyGoose You have to be a bit more specific: Do you mean condensed matter theory? If so, what exactly: General topics? Many-Body Theory?Spectroscopy? Solid state physics? --Let me know and I'll check what I know
Altland and Simon (Condensed Matter Field Theory). Chaikin and Lubensky (Principles of condensed matter physics), but this one I only skimmed through for some certain chapters. The books by Nozières and/or Pines are classic. A solid solid state book is Grosso Parravicini, and I think Ashcroft Mermin is fine too and covers many topics.
These are the first which come into my mind, but I will think about it a bit more. As I mentioned "condensed matter" is a very huge field, and with many sub-disciplines...
@SillyGoose "Berry phases in electronic structure theory" by Vanderbilt is a classic, too -- if you are interested in these kinds of things. IIRC, it should be available for free (the pdf).
What are some good condensed matter physics books that can fill the gap between Ashcroft & Mermin and research papers? Suggestions for any specialized topics (such as superconductivity, CFT, topological insulators) are welcomed.
I understand that nowadays nobody cares about Landau poles but I want to know why it was even considered serious historically? I mean, we find the one loop beta function under the assumption that higher loops are suppressed i.e. the coupling is small and then suddenly we are in awe that the coupling turns infinite at a finite energy?!
@NairitSahoo With one-loop you get - as you know - a $\log$ in the denominator
With higher loops resummation would yield $\log\log E$ terms
So the resummations turns out to be actually useful in the regime in which only the leading term matters, i.e. when $\log(E/\mu)\gg1$, having called $E$ the energy scale and $\mu$ the renormalization scale (e.g. the mass)
Once you accept that the addition of higher loop to the mix doesn't affect the qualitative behaviour at high energies, you should see why we are concerned with a divergent prediction: the coupling constant eventually gets so big that perturbation theory is meaningless, so you can't trust a calculation which uses the beta function produced by perturbation theory
Where have I mentioned that I want free phonons? :P
Maybe I have used improper terminology
I mean: when you write the ion-ion potential and expand wrt equilibrium positions, the zeroth order term is the static potential, the first order term vanishes and the second order term is the term that gives rise to free phonons
You can decide to quantize the third order terms and treat it perturbatively, though.
No, it is that usually you don't have phonon-phonon interactions, but rather you have electron-phonon interactions, which can then produce the phonon-phonon interactions ALA QFT
Mhh, how so? Isn't it a perturbation of the phonon hamiltonian?
You would get a $\sim (a_k+a_{-k}^\dagger)^3$ (where the power is a lazy way to mean factors with different momenta and $a$ is the destruction operator of phonons
This was briefly done in a lecture and they later stated it could be used to compute the phonon-phonon self-energy (you get diagrams like the gluon sunsets of QCD), but we did nothing excplicitly
Yes, if you have anharmonicity, then just as you have the horrible renormalisation in QFT, you would have to do that for the phonons and obtain an effective field theory
@Mr.Feynman The electron-nucleus potential is very different from the nucleus-nucleus potential. The Born-Oppenheimer approximation says that you fix the nuclei, so that the electrons feel attracted to the nuclei. But even here, it is not the full Coulomb potential, but usually a pseudo-potential that takes the repulsion of the core electrons into account. Then the nuclear-nuclear potential is the leftover attraction after the electrons have done their thing.
@Mr.Feynman You said that the perturbation theory breaks down at $\lambda \to \infty$. My objection is that it is obvious that it does, because we are not even considering the higher order loops to make these conclusions
How does people even say that a theory is strongly coupled or weakly coupled in IR or UV if they do not calculate the higher order beta functions?
(Okay, I sound like I am against the traditional mainstream viewpoint: but I am not... I am trying to understand how people makes these conclusions)
@NairitSahoo I don't understand what argument you think you're making here: One can, at any order, perfectly well imagine a running coupling that doesn't become infinite - and indeed Landau poles are absent from the Standard Model!
@ACuriousMind I am considering theories which have Landau poles from the beginning. In such theories why was Landau poles a big deal if we naively take the infinite coupling limit which we are not allowed to begin with because we are using one loop beta function and not the exact one?
One response to Landau poles was indeed that because they were derived by the very perturbation theory whose breakdown they signaled, they weren't really more concerning than just a large but finite coupling
alas, later non-perturbative methods confirmed that Landau poles signaled a problem with the theory at high energies at least in some cases - the now-known triviality of $\phi^4$ in the continuum limit of lattice models is a reflection of Landau's "renormalized charge going to zero" idea of resolving the pole
I am not familiar with every branch of physics out there, some of you have more ground covered, so here the question.
When we describe stuff like electrons or EM as waves. Do we really mean, these are waves, their nature and "true" self is a wave, or is the wave just a mathematical model, that we use to predict their behavior. does this question even make sense?
@Madder this question is essentially the question of "scientific realism" vs "anti-realism" and much in between, roughly speaking. I don't think there is a clear answer, since the answer depends on how you "see the world".
@TobiasFünke If you are telling me , that there is no answer, then for me, it is an answer. Also thank you for mentioning these terms, since i was not familiar with them.
@ACuriousMind The moment i sent the message, i asked myself the same question!
Then i thought of this!
Some mathematical models, could paint a false picture, but make true predictions. So if i imagine a ball as a mathematical manifold. Then i am not wrong, but at the same time, i am not right. It is not that the concept of a "manifold" something intrinsic or known to the cosmos. It is just there. (yes i am awre of space time manifolds)
check also "underdetermination of theory" in this context...in general, any good intro book on philosophy of science should cover all of this questions at least in a broad sense
I have one question about parity in general. If a function has negative parity, does it mean that the integral is zero? Or an answer cannot be given unless the boundaries of integrations are set?
As natural scientist/physicist, you are interested in creating models which correctly describes/predicts/explains certain (natural) phenomena and experiments. Whether or not the concepts of your model/theory are "real", is, as mentioned, to a very large extent a matter of personal beliefs/interpretation. Physics itself does not tell you if waves "are real" or that the gravitational force is "real" etc...
@imbAF How could the question possibly have an answer without knowing the bounds of integration? A continuous function for which all integrals are zero is zero itself.
@TobiasFünke Yea sure i agree with that, since there seems to be no objective way to determine that. But i was not sure, since no one really mentions those details.
@ACuriousMind I think in every sensible context (?), after all you're condensing the same observables into less degrees of freedom. You are losing something
@Mr.Feynman I'm not an expert on RG so I'm hedging my bets :P
@imbAF I suspect you have once again misunderstood the context. People tend to just write $\int f\mathrm{d}x$ without explicit bounds when the bounds are supposed to be clear from context (often the integral is then "over the entire space", whatever the "space" in the context is).
@ACuriousMind No space in particular is mentioned. You can argue, that because this is related to condensed matter theory, then the space is that of a crystal, which can be 1D etc. But no specific claim is made about it
I am pretty positive about this
But I want to show you that that is the case. Unless I can't see it
@ACuriousMind I want to know if it's the case. That I misunderstood the first part of the exercise
I think the sub exercise is pretty general in what is saying and asking.
It was more about calculating $\langle \psi |H| \psi \rangle$ , where $|\psi\rangle=\sum_\nu c_\nu |\nu\rangle$ ($|\nu\rangle$) eigenstates of H. In the end I got $\langle \psi |H| \psi \rangle=\sum_\nu |c_\nu|^2 E_\nu$. But I got confused a bit
i would like to know exactly what sort of calculations are done to analyze the data that something like CERN produces :P. it seems impossible to just use textbook QFT to model tons of uncontrolled scattering and decay events.
@Mr.Feynman when i say useful i mean useful for learning more physics or broadening one's view of physics heh
@SillyGoose Why would that seem impossible? You have to think in terms of an inexhaustible resource of grad students who can do annoying calculations :P
the assumption is that because from the viewpoint of the particles the beams are still kind-of sparse, you just compute the scattering results for one particle colliding with one other (or perhaps sometimes two or three or whatever) and then you just multiply by the luminosity of the beam
I don't mean just from agreement with experimental data. i mean from a thinking the assumption through sort of perspective. because one can make simplifying assumptions that capture broad strokes of what is going on while just being physically unreasonable.
like is the fact that these are high energy beams some sort of justification for this assumption? if we do such scattering at lower energies can we see a breakdown and etc.?
@SillyGoose I don't see an obvious reason to doubt it - the time scales on which these collisions happen are extremely short and at extremely small distances (the particles really have to hit each other almost head on), if you do some back-of-the-envelope calculation for the probability that when one particle hits another there's a third within relevant distance for the typical density of these beams, you should find something very small
is there a sort of fundamental theorem for variations? if i say that $\delta \int dr f(r) = f(r)$, then im able to make sense of a given theorem, but im not sure this is right
@TobiasFünke None. I follow the notes given by the professor. Which are taken from various books Peskin, Schwarz etc. But my plan is to read entire Schwarz once the current semester is over
What are some good condensed matter physics books that can fill the gap between Ashcroft & Mermin and research papers? Suggestions for any specialized topics (such as superconductivity, CFT, topological insulators) are welcomed.
