2329:! Please note, as is noted above, neither Kittel nor A&M nor any other standard CM textbook mentions "Bloch waves", but rather Bloch states and functions and electrons, etc. There is actually more confusion than that. I started a literature survey to try to pin down what they say, but some of the definitions are contradictory. Especially, for instance, that they don't have to be eigenstates - that depends on what book you're reading. The best route forward, that I see, is adding a "definition" section that tries to lay all that info out. Otherwise, I am a little confused about your statement because the introduction does not say that Bloch's theorem is the same thing as Bloch states. In fact, it defines Bloch's theorem as stating that the solutions of Schrodinger's equation in a crystal are given by Bloch states, which is supported by the literature (it does say that Bloch electrons are also called Bloch waves, which is not true for many reasons). Are you trying to argue they should have separate pages? If so, I don't think we have enough information to fill out a second article, and if we did we would have to decide on a title supported by the references, which is difficult as they all use different terms for some of the same concepts and they use the same terms for other concepts in which they all slightly differ. That is a primary reason the page is now titled "Bloch's theorem", as it is unambiguous, used in all the textbooks, and has the same definition everywhere (not to mention it is manifestly more notable, being mentioned every time any of the other terms are used). Let me know if I have misunderstood anything or if you have any other questions.
2349:! thanks for the comment. I indeed did not make the distinction between waves, wave functions, and states in the framework of quantum mechanics. I know there is a difference, but I do not think it is important for the current discussion. The question I want to discuss is if a Bloch wave-function is necessarily an eigen wave-function of the Hamiltonian, or if it can be more general defined. (Equivalent, is a Bloch state necessarily and eigen state of the Hamiltonian). The second sentence of the article now states that they have to be equivalent. Looking at the citations from literature above it is not used by these authors in quite that strict form. (Kitel contradict himself, in two different books A&M evade the pure definition by talking about Bloch electrons and Bloch form for the states or wave functions they encounter, Yu and Cardona state it does not need to be an eigen function.) Note that we all agree that eigen functions in a periodic solid are Bloch functions, but not all authors would state that a Bloch function is necessarily an eigenfunction of a periodic Hamiltonian.
2137:, sorry, I didn't realize you were going to respond. If I may take the moment, I am sorry for the previous comments. I was actually mistaken and feel pretty ridiculous about it. (For some reason I started to confuse wave-function and wave-equation. I guess the wave=wavefunction didn't sit right and I was otherwise just trying to justify it) I agree, though, that "Bloch wave" is seldom used and it might be better to use "Bloch theorem". Your proposed intro has my blessing, though I would probably want to work "Bloch electron" in there. The first sentence is spot on, I don't think we could do much better. You can also say the Bloch function is also called "Bloch state" and "Bloch wavefunction", but I don't think they have to be energy eigenstates in general, the literature is ambiguous on that, but they might use different terms. Otherwise we'll have to decide on how much information is really necessary, as you can always say that a state of that form can be written as a sum of energy eignestates, as they represent a complete basis set.
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waves is the crystal momentum of the eigen state. A Bloch wave is any wave that can be written on this form, not necessarily also an eigen state. (This definition is in agreement with both Kitel and A&M, although Kitel on page 180 might lead to confusion). Bloch Waves are used to calculate the eigen states in a solid. One starts with a set of atom centred functions. From those one can generate periodic functions by repeating the local function in each unit cell. Multiply with a plane wave generates a Bloch wave. Linear combinations of the Bloch Waves are Eigen-states in the periodic potential. Otherwise thanks for the good info!
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pp==800 which refers to a few like: density of states pp=143-146, dynamics pp=214-241, holes pp=225-229) referenced in the second part in a decent neat way, what is the distinction between an electron and a wave in any case ? As discussed in august Bloch waves is more of an historical term and is best off to be buried in a corner. The major prob of the discussion in august to me is the fact there was no Bloch theorem page which instead is the best recognizable term in recent years.
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wave-function, all others describe only form. Kittel only refers to the wavefunction being a Bloch wave function, not the Bloch wave being a Bloch wave-function. As you will find, no literature will ever support calling a wave and wave function the same thing, as it is mathematically incoherent. I will not revert your reversion, but I was not wrong for fixing the ambiguity.
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1233:, but the definition in the lead in not supported by the source. The definition I've gone back to reflects the survey of the literature done by the guy above. In your edit summary, you seem to be saying a Bloch wave and a Bloch wavefunction are different things? Kittel treats them as the same: "A one-electron wavefunction of the form (7) is called a Bloch function..." (
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multiplied with a plane wave, but are not eigen functions yet. These functions "serve as a suitable basis for the wave functions of electrons in crystalline solids." as they decouple wave functions with different crystal momentum. The statement that the eigen functions form a suitable basis as now made in the second sentence of this article is true but trivial.
1312:. Yes, I guess you could say I was trying to lay out a distinction between an abstract state in Hilbert space and its representation in position space, though from a less formal point of view, you could just say that a wave is a physical phenomenon, while a wave-function is any mathematical description of the wave. A wave obeys a wave-function.
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have the required translation property are known as Bloch waves. Since these wave functions..." It appears to hold Bloch waves and Bloch wave functions as synonymous (this is reinforced later when it says "The main point to remember is that each Bloch wave is associated with just one k(j) but it is a continuously varying function of r").
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term "potential part alone" comes from. In some texts they call them "electrons at rest" in a crystal which, although it may be "suggestive" of some magic that electrons can do, is even more confusing and is a plain heuristic narrative, this in fact refers to the fact that the stationary points of the wave don't actually move.
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I in fact disagree on the last distinction "potential part alone" in your book: an electron without kinetic energy does not make much sense, but some calls Bloch wave functions stationary waves and therefore they consider them with "total" and "average" kinetic energy zero, that is where probably the
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Tight-binding approximations are just a simple approximate expansion basis for the Bloch eigenfunctions in electronic systems. It doesn't have anything conceptually to do with Bloch's theorem per se. For example, you could just as easily use a tight-binding approximation in a non-periodic structure
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Okay, so I found a reliable source that talks about Bloch waves. When I first made the proposal above, I was new to
Knowledge, I've since gotten much better at looking for sources. This source, though is not a standard CM textbook, states that "The solutions to the Schrodinger equation which always
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Edit: I think the last sentence can be left off for the time being, and if we create the definitions section then a short summary of that section can go there with no need to double cite, it might could be read as OR the way it is now, I'm not sure. The content prior to that implies that it is more
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Dear all, I think there still is some confusion in the introduction. Bloch theory and Bloch Waves are two separate (but related) concepts. The Bloch theorem states that all eigenfunctions in a periodic lattice are of the form of a plane wave times a periodic function. The wave vector of these plane
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It's a good point that "Bloch wave" is actually fairly rare terminology. The most common/general terminology seems to be "Bloch's theorem": I can confirm it appears explicitly in Kittel Solid state, Ashcroft/Mermin, Griffiths QM, as cited above, and in several other books. As you mention, that does
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I see what you mean. I don't think anyone will object if you made that change, it sort of avoids the whole dispute. Like I said, different textbooks treat them differently, especially "Bloch states", as you can imagine, a Bloch state is almost always going to be an eigenstate as there is not much
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As you can see from the survey above, none of the authors of these texts support referring to the wave and wave-function as the same thing. They all are giving definition of the wave function by its mathematical form. The Bloch electron in
Ashcroft and Mermin is simply an electron that obeys this
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First Thanks for the comments, I think we shall promote any genuine contributor. Then for text improvements please go ahead, for major content improvements please add here in this talk a requirement section / requested content and then we can discuss it further or just type in here some proposed
1094:(they talk about "stationary states of Bloch electrons". Later in the book (page 185 and 215) they mention "Bloch wave" and "Bloch wave function" but in a way where it's ambiguous whether or not the term only refers to stationary states. Oh, and they use the term "Bloch form" as synonymous with
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I think the article splits the definition of Bloch's waves (which are always called Bloch's functions on purpose to avoid confusion and mention once to "less often called Bloch's waves") referenced in the first part from bloch electrons (i.e. electrons in a band - e.g. see
Ashcroft topic index
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has source code, but I'm very confused what you think the problem is. Are you referring to the fact that the bottom-left plot and bottom-right plot are the same curves? There's supposed to be the same curves. That's the whole point. Or are you talking about something else? Can you explain more
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The reason for using Bloch functions as not being eigen functions is that if one starts from localised orbitals (atomic like or
Wannier functions) and wants to create eigen states of the periodic lattice one in an intermediate step encounters functions that have the form of a periodic function
1391:. In order to discuss the distinction on Knowledge, we need to find a source that explicitly discusses such a distinction in this context. Apologies if the policy seems strange; the standards for writing a Knowledge article are surprisingly different than those for writing academic literature.
3165:. I have removed the vast majority of the backlinks to Bloch wave (most of which were really to Bloch wave-function, but "function" was left outside the link). If it is to be insisted that we keep it, it should go with the other names as it is clearly not a synonym for Bloch electron. Thanks!
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extra content and then we promote it to the page. (e.g. maybe it goes in other articles). In regards to literature search I would close off with this "Bloch Wave definition" discussion and move on to more interesting topics (i.e. not just definitions), e.g. spend effort on adding new content.
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In terms of references for the definition of Bloch Waves I would not use a specialist text book but a major undergrad / grad book, therefore missing a definition of Bloch's waves in a major such book I used instead the definition of the opposite term "Bloch
Electrons" as you see in
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We should use whatever definition is more common, according to popular textbooks and other sources. I believe it's the first one. But maybe I'm remembering wrong!! I don't have any books with me right this second. Let's work together to figure out which definition is more standard!
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No problem. It was a little confusing to understand what you meant, but you brought up some good ideas. Those kinds of tweaks sound reasonable enough. I've formally proposed the move in a new section below. Can you weigh in there? (see these discussions for examples:
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all list "Bloch's theorem" in their text and also list it in their back indices, whereas none of them use the term "Bloch wave" at all. It's also much less ambiguous on the precise meaning. (Do Bloch waves mean only eigenstates or also combinations of eigenstates?)
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Thanks for the heads up. I still think eventually the article should probably move "Theorem" to "theorem", but for now I'm more interested on working on some other articles. Feel free to make your own improvements. I keep an eye on this article, but I'm not here to
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I agree physical wavevectors with a.b = 2π δ_ij is the better convention. I'm just not sure the exponentials now have the correct factors. I said I would check tonight, but I'm tired from a very long day at work. I'll check them all tomorrow to make sure.
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For the record, I agree you are correct on the technical point, even if I don't think such a fine distinction is useful for a basic condensed matter physics article. But it doesn't matter what I think, or even what is true. Knowledge is
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I added e and i at the top. I don't see any reason to link the greek letters. The fact that ψ is called "psi" not "funny-looking greek letter" is a good thing to know but it's off-topic when discussing Bloch waves. (Just my opinion.)
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Maybe some improvement on the text of the article can help to clarify that Bloch's wave functions are actually eigenstates i.e. with just one k(j), eigenstates are mentioned a few times but maybe the text can be improved.
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My best guess is that you're trying to lay out a rigorous distinction between an abstract state in
Hilbert space and its representation in position space, but that is far too technical for this topic. We should write
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Stevenj: I think 134.99.136.28 was correct in capitalizing the nouns in the title of Bloch's paper: "Über die
Quantenmechanik der Elektronen in Kristallgittern". Nouns are written with capital letters in German.
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It seems to me that the last pair of images, in the figure comparing two expressions of the same Bloch state, shows two identical decompositions. The same image must have been taken twice by mistake !
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The derivation of the Bloch theorem is added. I am a new wikipedian and not very proficient in the wiki markup language and the formula is not very satisfactory. Could anyone help with that? Thank you
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Technical points: In the proposal template, I've mistakenly written "Bloch's
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point talking about it otherwise; the Bloch function, Bloch wavefunction, and "Bloch form" are all ambiguous. I think a definitions section will help clear up any other misunderstandings.
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The original derivation failed to properly divide k by 2pi, because the normalization convention for the reciprocal lattice vectors is that a_i \dot b_j = 2 \pi \delta_{ij}. I fixed it.
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I've learned that there are s,p and d bloch waves in transition metals. And that the rules for dipole radiation are very similar to free atoms with s,p,d orbitals. How is the connection?
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I agree with 68.55.253.191 that it's better to use a_i \dot b_j = 2 \pi \delta_{ij}, because the whole article is generally using angular wavevectors not normal wavevectors. I changed it
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as the potential part alone, not the entire wavefunction including the periodic potential and plane wave. I believe this calls for a further, more detailed survey of the literature.
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This follows loosely follows Kittel, without defining "Bloch wavefunction" since the term doesn't appear there. It still could be improved, but I wonder what you think of this.
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Please correct me if I'm wrong, but both users were referring to the use of the same plot in the bottom-left and bottom-right of that figure. Reinforcing what was said by
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do not use the terms "Bloch wave" or "Bloch function" or "Bloch wavefunction" in the main discussion of Bloch's theorem (chapter 8) as far as I see. They
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Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a
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Hi all, I will spend some time looking over sources to see what we can do about enhancing this article. I do, however want to point out
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per nominator. Bloch's theorem is regularly discussed in the literature and is an appropriate name for an overview of the topic.
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which suggests the first definition above, i.e. it suggests that anything called a "Bloch function" must be a stationary state.
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1247:. But maybe you mean something else? It would be helpful if you could point out some sources that support this distinction.
1151:(3rd edition) describes Bloch's theorem but doesn't use the terms "Bloch wave" or "Bloch function" or "Bloch wavefunction".
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Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
2221:, should we replace the introduction with the one you prepared? No rush, but I wanted to ask in case you forgot about it.
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is a propagating dynamic disturbance (change from equilibrium) of one or more quantities, sometimes as described by a
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to check if he is willing either to fix the image or to share the dataset to rebuild the graph. comment added by
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after discussing it on the closer's talk page. No further edits should be made to this discussion.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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A Bloch state is any state that is both (1) an energy eigenstate, and (2) can be written as
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It seems to me, we should use the second definition: A Bloch wave is any wave of the form
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This article is about quantum mechanics. For the theorem of complex analysis, see
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2159:) If there's no objection we can request a technical move in a week or so.
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Small mistake : wrong image in the "Applications and consequences" section
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mean rewriting the lead. So let's talk about it. How about something like:
699:'s change for now, pending this investigation. :-D Thanks in advance!! --
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where u has the same periodicity as the atomic structure of the crystal.
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since the theorem concerning crystals is roughly ten times as popular
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Transmission Electron Microscopy: A Textbook for Materials Science
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general and I think the details can be relegated to the body.
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it is, but again I could be wrong), then we need to change it
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Shouln't the Greek letter epsilon (ε) be used instead of "Є"?
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1808:(each has a different periodic component
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1745:
1741:
1735:
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987:
956:
951:
921:
905:
883:
877:
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841:
782:
763:
741:
735:
693:consistently throughout the whole article
656:
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635:
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629:
624:
612:
604:
560:
547:
539:
534:
533:
528:
516:
508:
3265:, Boston, MA: Springer US, pp. 235–243,
3242:
2995:
2902:2600:6C58:4B80:2727:6463:9C11:4331:9178
1524:for the wave functions of electrons in
60:
19:
2357:I would suggest a change of the form:
1512:. These solutions, sometimes known as
214:where Bloch's theorem does not apply.
1927:{\displaystyle \psi _{n\mathbf {k} }}
1873:{\displaystyle \psi _{n\mathbf {k} }}
1755:{\displaystyle \psi _{n\mathbf {k} }}
1350:is a mathematical description of the
818:which is the second definition above.
7:
2766:The result of the move request was:
2618:Functions of this form are known as
2380:. Mathematically, they are written:
106:This article is within the scope of
3035:Introduction to Solid State Physics
3008:Introduction to Solid State Physics
2928:Bloch's theorem (complex variables)
2921:Bloch's theorem (complex variables)
2813:Introduction to Solid State Physics
1528:. Mathematically, they are written:
1520:in energy, and serve as a suitable
1235:Introduciton To Solid State Physics
730:"A function of the form (19) [i.e.
203:? Would this improve this article?
49:It is of interest to the following
2923:can be accomplished by a hatnote,
2597:
2571:
2418:
2412:
1934:, is unique only up to a constant
1828:). Within a band (i.e., for fixed
1674:
1648:
713:UPDATE: Here are some references.
631:
625:
535:
529:
14:
3058:The discussion above is closed.
2805:Introduction to Quantum Mechanics
3376:Knowledge level-5 vital articles
2545:
2472:
2444:
2431:
2423:
2400:
2236:
2030:
2003:
2000:
1997:
1976:
1946:
1918:
1888:
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1794:
1782:is a discrete index, called the
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1622:
1553:
1457:(Reply to a now deleted comment)
657:
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613:
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517:
93:
83:
62:
29:
20:
3406:Mid-importance physics articles
3153:Bloch waves neq Bloch electrons
2044:can be restricted to the first
146:This article has been rated as
3386:B-Class level-5 vital articles
2891:. Further reference Sakurai's
2448:
2440:
2404:
2396:
2300:22:00, 11 September 2020 (UTC)
2231:07:36, 11 September 2020 (UTC)
2007:
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474:01:07, 21 September 2013 (UTC)
433:17:40, 20 September 2013 (UTC)
407:11:15, 20 September 2013 (UTC)
375:11:06, 20 September 2013 (UTC)
280:22:55, 19 September 2013 (UTC)
1:
3341:The graph with the caption,
3331:07:43, 24 December 2020 (UTC)
3314:07:43, 24 December 2020 (UTC)
3214:07:43, 24 December 2020 (UTC)
3193:07:43, 24 December 2020 (UTC)
2789:00:21, 5 September 2020 (UTC)
2740:Requested move 28 August 2020
2734:03:33, 20 February 2014 (UTC)
2716:22:41, 19 February 2014 (UTC)
2678:08:25, 24 November 2020 (UTC)
2660:08:19, 24 November 2020 (UTC)
2644:07:33, 24 November 2020 (UTC)
2368:states that solutions to the
2339:21:59, 23 November 2020 (UTC)
2320:21:14, 23 November 2020 (UTC)
2022:. Therefore, the wave vector
1500:states that solutions to the
189:19:38, 30 December 2005 (UTC)
179:13:21, 27 December 2005 (UTC)
126:Knowledge:WikiProject Physics
120:and see a list of open tasks.
3271:10.1007/978-0-387-76501-3_14
3233:21:30, 28 October 2020 (UTC)
3175:05:23, 28 October 2020 (UTC)
3145:16:42, 27 October 2020 (UTC)
3123:13:35, 27 October 2020 (UTC)
2688:e, i and r are not defined.
2604:{\displaystyle \mathrm {i} }
2578:{\displaystyle \mathrm {e} }
2552:{\displaystyle \mathbf {k} }
2479:{\displaystyle \mathbf {r} }
2037:{\displaystyle \mathbf {k} }
1953:{\displaystyle \mathbf {K} }
1902:, as does its energy. Also,
1895:{\displaystyle \mathbf {k} }
1801:{\displaystyle \mathbf {k} }
1681:{\displaystyle \mathrm {i} }
1655:{\displaystyle \mathrm {e} }
1629:{\displaystyle \mathbf {k} }
1560:{\displaystyle \mathbf {r} }
1218:19:14, 14 January 2014 (UTC)
721:Intro to Solid-State Physics
709:01:49, 13 January 2014 (UTC)
255:14:12, 3 December 2010 (UTC)
219:05:05, 27 January 2007 (UTC)
208:20:24, 26 January 2007 (UTC)
129:Template:WikiProject Physics
3357:17:22, 25 August 2022 (UTC)
3088:08:29, 2 October 2020 (UTC)
2986:21:14, 31 August 2020 (UTC)
2910:02:29, 31 August 2020 (UTC)
2884:23:38, 28 August 2020 (UTC)
2866:22:00, 28 August 2020 (UTC)
2209:22:30, 28 August 2020 (UTC)
2147:08:25, 28 August 2020 (UTC)
2119:07:33, 28 August 2020 (UTC)
1441:03:37, 27 August 2020 (UTC)
1374:02:19, 27 August 2020 (UTC)
1297:00:42, 27 August 2020 (UTC)
1201:{\displaystyle e^{ikx}u(x)}
1132:{\displaystyle e^{ikx}u(x)}
184:Ah, I didn't realize that.
3427:
2925:
2626:, and serve as a suitable
2050:without loss of generality
2048:of the reciprocal lattice
1711:electronic band structures
1381:not designed to be correct
1139:(page 139). So they don't
486:Definition of "Bloch wave"
152:project's importance scale
3337:The first graph is wrong?
3161:are not sometimes called
1880:varies continuously with
145:
78:
57:
3401:B-Class physics articles
3113:specifically? Thanks. --
3060:Please do not modify it.
2978:{\displaystyle \rangle }
2949:{\displaystyle \langle }
2893:Modern Quantum Mechanics
2858:{\displaystyle \rangle }
2829:{\displaystyle \langle }
2807:, Ashcroft and Mermin's
2754:Please do not modify it.
2362:condensed matter physics
2292:{\displaystyle \rangle }
2263:{\displaystyle \langle }
2201:{\displaystyle \rangle }
2172:{\displaystyle \langle }
2111:{\displaystyle \rangle }
2082:{\displaystyle \langle }
1494:condensed matter physics
1433:{\displaystyle \rangle }
1404:{\displaystyle \langle }
1327:, and related fields, a
1289:{\displaystyle \rangle }
1260:{\displaystyle \langle }
865:{\displaystyle u_{k}(x)}
827:Quantum Theory of Solids
169:Regarding capitalization
2811:, and Charles Kittel's
2561:crystal momentum vector
1638:crystal momentum vector
946:satisfies the equation
3371:B-Class vital articles
3108:The image detail page
2979:
2950:
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1237:, 8th edition, pp.167)
1202:
1133:
1066:
940:
872:of the Bloch function
866:
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668:
572:
3110:File:BlochWaves1D.svg
2980:
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2499:{\displaystyle \psi }
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1580:{\displaystyle \psi }
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36:level-5 vital article
3093:Dataset of the image
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2665:Maurits W. Haverkort
2636:Maurits W. Haverkort
2593:
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2370:Schrödinger equation
2327:Maurits W. Haverkort
2312:Maurits W. Haverkort
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260:Corrected 2 pi error
3037:. New York: Wiley.
3010:. New York: Wiley.
2809:Solid State Physics
1587:is the Bloch wave,
1084:Ashcroft and Mermin
201:Tight binding model
109:WikiProject Physics
3251:Williams, David B.
2975:
2946:
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2826:
2632:crystalline solids
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2012:
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1936:reciprocal lattice
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1526:crystalline solids
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568:
216:—Steven G. Johnson
186:—Steven G. Johnson
45:content assessment
3279:978-0-387-76501-3
2912:
2900:comment added by
2777:
2774:non-admin closure
2719:
2702:comment added by
2532:periodic function
2523:{\displaystyle u}
2378:periodic function
1841:{\displaystyle n}
1821:{\displaystyle u}
1775:{\displaystyle n}
1609:periodic function
1600:{\displaystyle u}
1510:periodic function
969:
492:reverted a change
270:comment added by
166:
165:
162:
161:
158:
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3418:
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3255:Carter, C. Barry
3247:
3221:Bloch's function
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1092:stationary state
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807:] is known as a
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3159:Bloch electrons
3155:
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3031:Kittel, Charles
3029:
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3004:Kittel, Charles
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2800:Bloch's Theorem
2798:
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2620:Bloch functions
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2376:modulated by a
2366:Bloch's theorem
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1967:
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1514:Bloch functions
1508:modulated by a
1498:Bloch's theorem
1422:
1421:
1415:
1393:
1392:
1354:of an isolated
1348:quantum physics
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243:
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131:
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125:
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43:on Knowledge's
40:
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12:
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5:
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3422:
3414:
3413:
3408:
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3398:
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3388:
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2032:
2009:
2005:
2002:
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1995:
1992:
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1984:
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1676:
1664:Euler's number
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1555:
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1356:quantum system
1336:
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1245:one level down
1239:
1238:
1197:
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1157:Yu and Cardona
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836:"The function
831:
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813:
812:
809:Bloch function
796:
793:
790:
785:
781:
775:
772:
769:
766:
762:
758:
755:
752:
749:
744:
740:
725:
724:
681:
680:
679:
678:
677:
676:
663:
659:
655:
652:
646:
642:
638:
633:
627:
622:
619:
615:
611:
608:
585:
584:
583:
582:
581:
580:
567:
563:
559:
556:
550:
546:
542:
537:
531:
526:
523:
519:
515:
512:
487:
484:
483:
482:
481:
480:
479:
478:
477:
476:
440:
439:
438:
437:
436:
435:
412:
411:
410:
409:
378:
377:
338:
329:
320:
311:
304:
297:
290:
261:
258:
242:
241:The derivation
239:
234:
231:
226:
223:
222:
221:
196:
193:
192:
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170:
167:
164:
163:
160:
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148:Mid-importance
144:
138:
137:
135:
118:the discussion
105:
104:
101:Physics portal
88:
76:
75:
73:Mid‑importance
67:
55:
54:
48:
26:
13:
10:
9:
6:
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3:
2:
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3389:
3387:
3384:
3382:
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3377:
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3300:
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3276:
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3268:
3264:
3260:
3259:"Bloch Waves"
3256:
3252:
3246:
3243:
3239:
3234:
3230:
3226:
3225:Footlessmouse
3222:
3215:
3211:
3207:
3202:
3201:
3200:
3194:
3190:
3186:
3183:
3179:
3178:
3177:
3176:
3172:
3168:
3167:Footlessmouse
3164:
3160:
3152:
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3142:
3138:
3137:Footlessmouse
3134:
3130:
3129:
3128:
3127:
3124:
3120:
3116:
3111:
3107:
3106:
3105:
3104:
3100:
3092:
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3043:0-471-14286-7
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2965:
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2922:
2917:
2911:
2907:
2903:
2899:
2894:
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2887:
2885:
2881:
2877:
2876:Footlessmouse
2873:
2870:
2869:
2868:
2867:
2845:
2841:
2814:
2810:
2806:
2801:
2796:
2791:
2790:
2787:
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2775:
2769:
2762:
2760:
2755:
2750:
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2727:
2722:
2721:
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2713:
2709:
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2701:
2683:
2679:
2675:
2671:
2670:Footlessmouse
2666:
2663:
2661:
2657:
2653:
2652:Footlessmouse
2648:
2647:
2646:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2616:
2614:
2588:
2562:
2537:
2533:
2517:
2509:
2508:wave function
2493:
2486:is position,
2461:
2437:
2427:
2407:
2393:
2385:
2381:
2379:
2375:
2371:
2367:
2363:
2358:
2355:
2348:
2347:Footlessmouse
2344:
2343:
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2336:
2332:
2331:Footlessmouse
2328:
2324:
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2184:
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2120:
2098:
2094:
2068:
2067:
2066:
2065:
2064:
2063:
2062:
2061:
2051:
2047:
1990:
1986:
1982:
1972:
1968:
1937:
1914:
1910:
1860:
1856:
1835:
1815:
1785:
1769:
1742:
1738:
1729:
1728:
1727:
1726:
1725:
1724:
1723:
1722:
1712:
1708:
1707:
1706:
1705:
1704:
1703:
1702:
1701:
1691:
1665:
1639:
1614:
1610:
1594:
1574:
1567:is position,
1544:
1543:
1542:
1541:
1540:
1539:
1538:
1537:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1499:
1495:
1491:
1490:
1489:
1488:
1487:
1486:
1485:
1484:
1473:
1472:
1471:
1470:
1469:
1468:
1467:
1466:
1456:
1455:
1454:
1453:
1452:
1451:
1450:
1449:
1442:
1420:
1416:
1390:
1386:
1382:
1377:
1376:
1375:
1371:
1367:
1366:Footlessmouse
1362:
1357:
1353:
1352:quantum state
1349:
1345:
1344:wave function
1341:
1340:Wave function
1337:
1334:
1333:wave equation
1330:
1326:
1322:
1318:
1314:
1313:
1311:
1304:
1303:
1302:
1301:
1298:
1276:
1272:
1246:
1241:
1240:
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1232:
1227:
1226:Footlessmouse
1222:
1221:
1220:
1219:
1215:
1211:
1192:
1186:
1181:
1178:
1175:
1171:
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1117:
1112:
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195:Tight binding
194:
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3323:Flyredeagle
3306:Flyredeagle
3206:Flyredeagle
3185:Flyredeagle
3163:Bloch waves
3103:flyredeagle
2759:move review
2536:wave vector
1613:wave vector
1518:eigenstates
1325:mathematics
697:User:Edib76
588:User:Edib76
3365:Categories
3285:2020-10-28
3238:References
3099:Sbyrnes321
3073:Dear all,
2991:References
2795:Bloch wave
2768:page moved
2374:plane wave
1784:band index
1506:plane wave
1387:, and you
3298:Ashcroft.
1149:Griffiths
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394:contribs
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176:O. Prytz
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465:physics
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2785:{talk}
2780:~SS49~
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