Connections between classical mechanics and quantum mechanics I've been studying quantum mechanics and classical mechanics for a little while now, and I still don't feel as though I fully understand the motivation for some of our choices in Heisenberg mechanics.  For example, it clearly isn't a coincidence that the classical observables (functions of coordinates and their conjugate momenta) and the quantum observables (Hermitian operators) seem to form analogous Lie algebras with the Poisson bracket and commutator respectively.  But it isn't clear to me why this is true.  Is there some deep meaning contained in this statement?  Or is it more indicative of the fact that in constructing a quantum model of the universe we took substantial inspiration from our intuition and previous study of classical mechanics?  Along these same lines, what motivates the move from classical functions on phase space to Hermitian operators?  I understand why operators corresponding to observables must be self-adjoint (the eigenvalues must be real), but I don't understand what motivates the move to operators in general.  Why would we expect that operators on a Hilbert space would give physical predictions?  Part of my confusion here may also come from the fact that it isn't entirely clear to me what exactly these operators do in all cases.  For example, I get that $\langle \psi | \hat{x} | \psi \rangle$ corresponds to the expected position of a particle in state $|\psi\rangle$, but it's much less obvious what the $\hat{x}$ operator does to a state in general.  In some cases (such as $J_\pm$ when considering angular momentum), it's clear what the operator does to a state (raises or lowers eigenstates of $J_z$), but in all these cases the operator is non-Hermitian.  Perhaps the answer to this question is simply that the model gives accurate predictions and so we use it, but I'm wondering if there's a better way to think about these things.
Why quantum mechanics? Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before.  How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?  If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious.  Why should this be the case, or at least, why might we expect this to be the case?   Then there is the issue of measurement which is even more difficult to motivate.  In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$).  How would you convince your students that this had to be the case?  I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be.  So, what do you guys think?  Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?  Note that, at some point, you will have to make reference to experiment.  After all, this is the reason why we needed to develop quantum mechanics.  In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying.  I think we can do better.  Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory.  You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.  Edit:  To clarify, I will try to make use of a much simpler example.  In an ideal gas, if you fix the volume, then the temperature is proportional to pressure.  So we may ask "Why?".  You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.".  If you choose the latter, you can then ask why that is true.  Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example).  The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment.  And in this particular case, the objective is to do this with quantum mechanics.
How to get the Infobox data from Wikipedia? How to install ubuntu on lenovo g580? I'd like to install Ubuntu 13.04 on Lenovo g580 and I have this issue. After a few seconds of booting liveUSB screens gets black and nothing happens. I've read somewhere about connecting second screen to solve similar issue. Of course this didn't solve it, but it provided something like terminal with     initramfs>   and few basic commands ..  Is it possible to install Ubuntu on Lenovo g580 hardware? I also checked  Ubuntu 12.04: failed  Mint 13: SUCCEED (x32 and x64)  Is it because of drivers? Of course I want Ubuntu instead of Mint. :) Why does demand for stock rise when a company appears to have high future value? So a company decides to raise money by selling shares of its stock. The stock is offered at an initial price of their determination. The stock is then traded in a market where forces of supply and demand, as well as speculation, affect its price.  When a company performs well (or makes decisions that will likely increase the company's earnings in the future), more people want to buy that stock than do people wanting to sell it. This creates demand for that stock which causes its sell/bid prices rise. When a company performs poorly, the opposite occurs; more people want to sell the stock than want to buy. This creates a surplus that causes sell/bid prices to fall.  Understanding this, people speculate over company performance to determine the future behavior of a stock, and attempt to "buy low, sell high" or "borrow high, return low" in order to make a profit.  But my question is regarding the first part of that process. What is it about company performance that causes the perceived value of its stock to rise? This is especially perplexing since I would guess that most people don't exercise their voting rights, and many stocks don't pay dividends.  It's not like a shareholder has any direct financial gains from increased company earnings (speculative stock pricing notwithstanding).  What is the intrinsic value of the stock?  It seems to me that all participants in a stock market tacitly agree that stock price will have a direct correlation with company performance. So another way of asking the question is, would an opposite agreement hold? What financially (this isn't a question about logistics, communication, or feasibility) stops all participants in a market from agreeing to value a particular company's stock inversely with respect to the company's performance?  Since I feel like I can't quite nail down my question, another form of it would be, why is stock market behavior driven by company performance? Set the maximum character length of a UITextField Upgrade or replace a Wubi installation? I've got Windows 7 with Wubi 11.10 installed. I would like to upgrade to 12.04 but I'm not sure what the best path would be.  I'm not skilled enough to use Ubuntu all the time, and my iPhone requires me to run some Windows version for the damn iTunes software. I would love to run Windows XP but I can't figure out how to install it via USB -- my computer has no cd drive.  I'd like to run Windows XP and Wubi 12.04 on top of that. Or, perhaps, Ubuntu 12.04 natively and Windows XP alongside of that. So I guess I have 2 questions:   (lazy) Should I upgrade my Wubi from inside Ubuntu, or should I remove Wubi 11.10 from within Windows and install a brand new Wubi 12.04? (proper) How can I install Windows XP and Ubuntu 12.04 alongside each other?   Update: I am going to create a separate question post about the above #2 question... I realize it's wrong to ask two questions in one post when they are that different. Show this equality (The factorial as an alternate sum with binomial coefficients). Spin or rotate an image on hover What is wrong with defining \bal as \begin{align}? I'm trying to define two commands \bal and \eal instead of inserting \begin{align} and \end{align} in my document, but in the code below, I'll get some errors. What is wrong with it?    \documentclass{article} \usepackage{amsmath} \makeatletter  \newcommand{\bal}{%  \begingroup  \align% \@ifstar{\nonumber}{}%  }  \newcommand{\eal}{\endalign\endgroup} \makeatother \begin{document} \bal* x=y \eal ‎\end{document} How to find and return a duplicate value in array How can I mirror one of the screens in a 3-monitor setup? Does an Animagus's animal form change along with their Patronus? In the two examples shown of an Animagus capable of casting a corporeal Patronus (Minerva McGonagall and James Potter) both had the same animal as their Patronus and as their Animagus animal.  Now, it is also shown that extreme circumstances can change one's Patronus. I think the only example of this was Tonks, which turned from a jackrabbit into a wolf in response for her love of Lupin.  If Tonks (or any other wizard who had their Patronus change) had been an animagus, would her animagus have changed as well? I would like to ask if MS Access database supports working in networks &amp; multi-user access? A way for "new users" to ask about their post specifically We currently require that new users have 5+ reputation to participate in meta support discussions (bugs, feedback, governance, etc). This is primarily because the use case for meta is not intrinsically obvious, so we require a modicum of "experience" with the site before jumping into it.   But that blocks new users with legitimate questions about their own posts.  Setting aside any arguments about new-users-are-evil-and-suck-the-life-out-of-our-community, I would like to propose a very specific path for new users to ask about their own question provisionally.  Instead of blocking a new user with this message:     You must have at least 5 reputation on &lt;site> to ask a question on meta.   We can change this to:     You must have at least 5 reputation on &lt;site> to post on meta, but you may ask a question about your own post specifically.      Link to your post: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;      ask about this post   cancel    The system will check whether the linked question provided belongs to the author. . The user can then proceed with the post provided they leave the template (below) intact:     Title &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;      &nbsp;         &lt;!--Do not remove-->   &lt;!--Note: Questions not about your post will be deleted-->      I have a question about the post:       &lt;!-- Your question below: -->   &nbsp;     Not only is this a great and gentle introduction to the proper use of meta, but it stands to help new users just when they need it most (learning to use the community).   Of course this doesn't 100% guarantee that someone won't get it wrong on occasion. But this is an awful lot of in-your-face guidance without simply telling people "read our FAQ" (which they never do). And if a user manages to step over all that guidance and still manages to misuse meta, I have no compunction about summarily removing a question they should never have posted.   But this just-in-time learning is a very effective way of learning a new site in a gentle and thoughtful way. If this is abused more often than not, it is easy to disable. But it's better than assuming clueless newbie every time. Coffeescript existential operator sometimes checks against undefined, but not always Moderator actions on items in the review queues don't count towards the review totals I am trying to run some unit tests in a C# Windows Forms application (Visual Studio 2005), and I get the following error:  System.IO.FileLoadException: Could not load file or assembly 'Utility, Version=1.2.0.200, Culture=neutral, PublicKeyToken=764d581291d764f7' or one of its dependencies. The located assembly's manifest definition does not match the assembly reference. (Exception from HRESULT: 0x80131040)** at x.Foo.FooGO() at x.Foo.Foo2(String groupName_) in Foo.cs:line 123 at x.Foo.UnitTests.FooTests.TestFoo() in FooTests.cs:line 98** System.IO.FileLoadException: Could not load file or assembly 'Utility, Version=1.2.0.203, Culture=neutral, PublicKeyToken=764d581291d764f7' or one of its dependencies. The located assembly's manifest definition does not match the assembly reference. (Exception from HRESULT: 0x80131040)  I look in my references, and I only have a reference to Utility version 1.2.0.203 (the other one is old). Any suggestions on how I figure out what is trying to reference this old version of this DLL file? Besides, I don't think I even have this old assembly on my hard drive. Is there any tool to search for this old versioned assembly? Sentence vs proposition Different ways to execute a shell script There are several ways to execute a script. The ones I know are: /path/to/script # using the path (absolute or relative) . script        # using the . (dot) source script   # using the `source` command  Are there any other way? What are the differences between them? Are there situations that I must use one and not another? What's the blue pentagon for in Pokemon XY? On the Pokemon status screen, underneath the Pokemon and to the left of the custom markings, is a small blue pentagon:  This doesn't seem to be present on Pokemon obtained through the Pokemon Transporter from Gen V:  I've looked into it myself, and popular opinion seems to be that it's either some kind of &quot;Legitimate Pokemon&quot; mark or simply to indicate that the Pokemon originates from Gen VI. The latter seems far more likely to me, but do we know for sure what this blue hexagon means? Where Is Shadeless Material Option For Blender 2.8? just trying to use shadeless lighting for a material, but cant find where it is in 2.8 This is how it looked in the older versions:    Any help is appreciated, thank you!