As far as I understand the postulates of quantum mechanics use only properties of abstract Hilbert space. So could we use any other Hilbert space for calculations instead of $L_2$? What could it be?  I can only suggest infinite-dimensional vector space with Euclidean norm, but it seems to be isomorphic to $L_2$.
I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space of state functions, from where this arises? and why 2-order space that assumes the following inner product:  $\langle\phi|\psi\rangle =\int\phi^{*}\psi\,dx$  While there is many ways to define the inner product.  In Physics books, this always assumed as given, never explains it, also I tried to read some abstract math books on this things, and found some concepts like "Metric weight" that will be minimized in such spaces, even so I don't really understand what is behind that, so why $L_2$? what special about them? Who and how physicists understood that those are the one we need to use?
Does anybody know which files I need to save in order to have a simple backup?   Reason for my question is, that my Galaxy Android smartphone lost all calendar data without any interaction. I found tons of apps, tons of sync stuff, but NO simple list of files to save. I do NOT want to save my data remotely on Google.  There must be at least one local file (database?) on my phone containing the synced calendar file (on my server, using ICSSync) I am new to Ubuntu, but I believe I have Ubuntu 11.10 installed correctly.  I am trying to install Acroread using AdbeRdr9.4.6-1_i386linux_enu.deb that I downloaded from the Adobe web site.  How do I do this? Display keyboard without animation How exactly does one “control for other variables”? Here is the article that motivated this question:   I liked this article, and it nicely demonstrates the concept of “controlling for other variables” (IQ, career, income, age, etc) in order to best isolate the true relationship between just the 2 variables in question.    Can you explain to me how you actually control for variables on a typical data set?      E.g., if you have 2 people with the same impatience level and BMI, but different incomes, how do you treat these data?  Do you categorize them into different subgroups that do have similar income, patience, and BMI?  But, eventually there are dozens of variables to control for (IQ, career, income, age, etc)  How do you then aggregate these (potentially) 100’s of subgroups?  In fact, I have a feeling this approach is barking up the wrong tree, now that I’ve verbalized it.  Thanks for shedding any light on something I've meant to get to the bottom of for a few years now...! Shine a light into a superconductor A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.  And what about type-II superconductors? Who is / was the "Lord of the Rings"? Who is the Lord, as referred to in the title? Frodo? Sauron? Someone else? Or is it more of an abstract title, not referring to a single person, like "He who wields the One Ring"? Why does String.split need pipe delimiter to be escaped? Probability of a 2 pair in poker Which characters are valid in CSS class names/selectors? What characters/symbols are allowed within the CSS class selectors? I know that the following characters are invalid, but what characters are valid?  ~ ! @ $ % ^ &amp; * ( ) + = , . / ' ; : " ? &gt; &lt; [ ] \ { } | ` # When programming interfaces, I've found I'm doing a lot of casting or object type conversion.  Is there a difference between these two methods of conversion?  If so, is there a cost difference or how does this affect my program?  public interface IMyInterface {     void AMethod(); }  public class MyClass : IMyInterface {     public void AMethod()     {        //Do work     }      // Other helper methods.... }  public class Implementation {     IMyInterface _MyObj;     MyClass _myCls1;     MyClass _myCls2;      public Implementation()     {         _MyObj = new MyClass();          // What is the difference here:         _myCls1 = (MyClass)_MyObj;         _myCls2 = (_MyObj as MyClass);     } }   Also, what is "in general" the preferred method? I have a device with a web-based control panel, and accidentally set it to redirect all http pages to https, even though some don't work over https. Although I've since corrected this, Safari seems to have memorised the redirect and is refusing to forget it, instead constantly attempting to redirect me to the invalid https address.  I've already closed Safari, cleared ~/Library/Caches/com.apple.Safari/ and ~/Library/Cookies/HSTS.plist but it still seems to be remembering the redirect when I re-open it.  Where else could Safari be storing this information? I can access the correct page via Firefox or Chrome, so it may not be a system wide service, or if it is it's not one that the other browsers use.  Unfortunately because the web-panel is provided by a device I don't believe I can adjust headers or setup a redirect back to the correct URL, which seem to be options offered in other similar questions, so I really need to find out where this data is being stored so I can destroy it with fire. Translate of an ideal by $a$ coincides with the ideal iff  $a \in I$ How  can I show the following?     Show that, if $a$ and $b$ are elements of a ring $R$ and $I$ is an ideal of $R$, then $$a+I=0+I \iff a \in I$$    I am so interested to know the proof.Thanks and good night all! How to install a QGIS plugin when offline? Due to various IT policies at my workplace, QGIS is installed on a machine that is not connected to the internet. I wish to install a couple of QGIS plugins on this system.  I have downloaded the required plugins from . How do I install them in QGIS? Why $f(x) = \sqrt{x}$ is a function? Zoom in/out on click in OpenLayers 3? %AX = (%AH + %AL)  So why not %EAX = (%SOME_REGISTER + %AX) for some register %SOME_REGISTER? Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane that is perpendicular to $\alpha$ the elements of  $\Phi$ in $V$.   I want to show that:   The Weyl chambers are open, convex and connected.   My attempts:   A single hyperplane $H_\alpha$ separates the space $V$ into two half-spaces: positive half-space $H_\alpha^+=\{x \in V \mid (\alpha,x)&gt;0\}$ and negative half-space $H_\alpha^-=\{x \in V \mid (\alpha,x)&lt;0\}$. It is clear that $V \setminus H_\alpha = H_\alpha^+\cup H^-_\alpha.$ As the result, the Weyl chambers can be regarded as components of  $$V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}= \bigcap_{\alpha \in \Phi} \left( V \setminus H_{\alpha} \right)= \bigcap_{\alpha \in \Phi} \left( H_\alpha^+\cup H_\alpha^- \right).$$ By the discussion above, a Weyl chamber is simply a $\textbf{intersection}$ of finite half-spaces. It is the case since one may write $$\left( H_\alpha^+\cup H_\alpha^- \right) \cap \left( H_\beta^+\cup H_\beta^- \right) =\left( H_\alpha^+\cap H_\beta^+ \right)  \cup\left( H_\alpha^+\cap H_\beta^- \right)  \cup \left( H_\alpha^-\cap H_\beta^+ \right)  \cup\left( H_\alpha^-\cap H_\beta^- \right).$$ Therefore openness and convexity of the Weyl chambers are straightforward since half-spaces are open and convex, so are their intersections. Moreover, we would replace $V$ by $\Bbb R^{\dim V}$ since they are isomorphic. Now we can prove that the Weyl chamber is connected by showing that every convex set in $\Bbb R^{\dim V}$ is connected. And I'm sure that proof of this statement existed. So am I done? Access nested dictionary items via a list of keys? I'm having some problems with listings and UTF-8 in my document. Maybe someone can help me? Some characters work, like é and ó, but á and others appear at the beginning of words...  \documentclass[12pt,a4paper]{scrbook} \KOMAoptions{twoside=false,open=any,chapterprefix=on,parskip=full,fontsize=14pt}  \usepackage[portuguese]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{listingsutf8}  \usepackage{inconsolata} \lstset{     language=bash, %% Troque para PHP, C, Java, etc... bash é o padrão     basicstyle=\ttfamily\small,     numberstyle=\footnotesize,     numbers=left,     backgroundcolor=\color{gray!10},     frame=single,     tabsize=2,     rulecolor=\color{black!30},     title=\lstname,     escapeinside={\%*}{*)},     breaklines=true,     breakatwhitespace=true,     framextopmargin=2pt,     framexbottommargin=2pt,     extendedchars=false,     inputencoding=utf8 }  \begin{document} \begin{lstlisting} &lt;?php  echo 'Olá mundo!'; print 'Olá mundo!'; \end{lstlisting}  \end{document} \end{lstlisting} This has happened to me many times in the last 5 years: an upgrade broke my system. Each time I end up with this situation, I have to reinstall the whole system, which is really annoying.  Is there any way to roll back the most recent upgrade to be able to have a functional system without reinstalling? If not, which is the best way to suggest this as a top priority idea?   I read this idea was described in brainstorm.ubuntu.com, but it feels it is dead... and the forums are full of examples of upgrades breaking things, that's why I feel something needs to be done about this topic. Thanks!