We know that quantum mechanics requires self-adjoint operators, not only symmetric. Can we say that this follows ONLY from the two following axioms of quantum mechanics, namely that    each observable $a$ corresponds to a linear operator $A$    and    an expectation value of a measurement of $a$ must be real   ?  I thought these two imply only the need for a Hermitian (i.e. symmetric) operators (because a linear Hermitian operator has real eigenvalues)  and that the need for self-adjointness was somehow connected to an additional requirement such as unitarity of the time-evolution operator. What is the missing piece?   (I know how the two terms are defined, e.g. .)
A symmetric operator has only real eigenvalues and different eigenvectors corresponding to different eigenvalues are orthogonal.   These are exactly what we want for a physical observable. I think this is the all required.   So why is quantum mechanics not satisfied with symmetric operators, but wants more?
Animations with movie15 and swf files I am trying to use movie15 to embed vector animations in my presentation. The trouble is that embedded SWF files refuse to display on Acrobat Reader X sometimes with a "This Media requires an additional player" message. Eg. this  doesn't work. Animated GIFs won't display either, with no error message.   Maybe this is an Acrobat issue but I tried the flashmovie package with the same SWF files and that does work (but only with the embedded player option so no controls are visible). But flashmovie is supposedly unstable with beamer so I was wondering if someone had run into the same problem with movie15.  I am using Windows 7, Acrobat 10.1.2 and MiKTeX. How do I make Chromium use Flash from Google Chrome? Now, that Adobe has stopped issuing new releases for Linux desktops, the only option for users is to use Google Chrome, for those who want to use the latest Flash updates. I use Chromium. Is there any way to get Chromium use the Flash from Google Chrome? The reason I as this question is that I have noticed both these browsers run in the background, so technically, the browsers' plugins are available to use. Proving a triangle with different edge colors exists in a graph. This is again some homework translated (hopefully not too badly) from my book  The graph $K_{n}$ is colored using $n$ different colors, in a way that each color is used at least once. Prove that there exists a triangle with its edges colored in different colors. I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post:   So I got the information, that the vector $|x\rangle$ form a dirac-normalized basis for the Hilbert Space.   I know that the dirac-delta distribution is defined like this: $$\delta(x-x') = \begin{cases} 0 &amp;\mbox{if } x\neq x' \\ \infty &amp; \mbox{if } x=x' \end{cases},$$ this means that my x' is a point on my x axis where I have my infinite high peak. And also $$\int_{-\infty}^{\infty}dx\cdot \delta(x-x')=1.$$  But how actually correlate this with the scalar product of vectors x, x' in the Hilbert Space that form a so-called 'diracl-normailzed" basis of it?  Can you give me some tips on this please? Or maybe you actually know a link, where this is explained. When disconnecting from WebSphere MQ with C# client TCP connections are still in CLOSE_WAIT status Non-principal ideal in $K[x,y]$? How to close/open a view panel I can't find a way to close a view panel, in particular the horizontal at the bottom, is pretty useless to me at this moment. Scripting `rm` and `rm -i` I want to create a script that whenever there are more than three files being removed, it invokes rm -i by default. I know it will go in initialization files but can't seem to get it working.  So far:  if [$file -ge 3]; then   rm -i    exit 0  else   rm exit 1 Can Cycles renders be done from the command line? It seems that Cycles is mainly built in the form of Python scripts (as far as I can tell). Is it still possible to use Cycles to render a .blend file from the command line? Why can't we see images reflected on a piece of paper? I dont have a clue about the proof in the reverse implication on the iff statement.  I will show what I have by now:  1) A cofinite topology implies that any infinite subset in $X$ is dense. Let $U\in\tau$ and is cofinite topology so $U=X-\bigcup_{k=1}^{n}\{x_k\}$.  Let $V\in X$ an infinite subset then $$V\cap U=V\cap(X-\bigcup_{k=1}^{n}\{x_k\})=(V\cap X)-(V\cap \bigcup_{k=1}^{n}\{x_k\})=V-(V\cap \bigcup_{k=1}^{n}\{x_k\})$$  and cause $|V|\ge\aleph_0$ and $|\bigcup_{k=1}^{n}\{x_k\}|=n\in\Bbb N\,$ we have that $|V\cap \bigcup_{k=1}^{n}\{x_k\}|\le n$. And finally is obvious that $V-(V\cap \bigcup_{k=1}^{n}\{x_k\})\ne\emptyset$ so any infinite subset is dense on $X$.  2) A cofinite topology implies a $T_1$-space. Let $x, y\in X$ and $U\in\tau: U=X-\{y\}$ so $x\in U\land y\notin U$. We can define too $V\in\tau:V=X-\{x\}$ so $x\notin V\land y\in V$ then a cofinite topology is a $T_1$-space.  But my problem is that I dont have a clue to prove the reverse implication i.e. if we have a $T_1$-space where every infinite subset is dense then it is a cofinite topology. Some hint? Some advice or correction in the proof of above? Thank you in advance.    I will complete the proof with the advice of @Cameron Buie.  1) A $T_1$-space implies that any finite subset is closed. Let the singleton $\{x\}\subset X$. Then $\overline{\{x\}}=\{x\}\cup\{x\}'$ where $\{x\}'=\{y:(\forall U\in\tau : y\in U)\land x\in U\}$, but because we are in a $T_1$-space exist some $U$ for any points $x,y\in X: x\in U\land y\notin U$ so $\{x\}'=\emptyset$ then $\overline{\{x\}}=\{x\}$ and is closed.  This means that for every $T_1$-space it cofinite sets are open.  2) If any infinite subset is dense on $X$ it means that the closure of any infinite subset is $X$ so the only closed sets are finite (and $X$ itself), so the topology is cofinite. How to delay a program such as cairo-dock (No openGL) or Firefox?  Sometimes programs crash when started with other startup apps. I want programs to start 10 seconds late to avoid a crash. If angular velocity & angular acceleration are vectors, why not angular displacement? Are angular quantities vector?   ... It is not easy to get used to representing angular quantities as vectors. We instinctively expect that something should be moving along the direction of a vector. That is not the case here. Instead, something(the rigid body) is rotating around the direction of the vector. In the world of pure rotation,a vector defines an axis of rotation, not a direction in which something moves.   This is what my book (authored by Halliday, Walker, Resnick) says. But they mentioned with a caution that      Angular displacements cannot be treated as vectors. ....to be represented as a vector, a quantity must also obey the rules of vector addition..... Angular displacements fail this test.   Why do they fail in this test? What is the reason? If I change the password for the root user is that going to automatically change the password for the user account?   I found this link for root:    How do I change the user account password? Can there be temperature inside of a massive Black Hole? In the Prisoner of Azkaban, Harry and Lupin could practice the patronus on the boggart, but when Harry was in the final task in the Goblet of Fire, he tried to use the patronus on the boggart, it didn't work and he then moved on to use ridikkilus.  Why could he use a patronus against the boggart, but then not in another? There is no such thing as magnetism? Here's an interesting "proof" that there is no such thing as magnetism. I know the answer but I love this so much I had to ask it here. It's a great way to confuse people!  As we all know, $$\nabla \cdot\vec{B} =0$$ Using the divergence theorem, we find $$ \iint_S \vec{B} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{B} \, dV = 0$$ Since $\vec{B}$ has zero divergence, there exist a vector function $\vec{A}$ such that $$\vec{B} = \nabla \times \vec{A}$$ Combining the last two equations, we get $$\iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Applying Stokes' theorem, we find $$\oint_C \vec{A} \cdot \hat{t} \, ds = \iint_S \hat{n} \cdot \nabla \times \vec{A} \, dS = 0$$ Therefore, $\vec{A}$ is path independent and we can write $\vec{A} = \nabla \psi$ for some scalar function $\psi$. Since the curl of the gradient of a function is zero, we arrive at: $$\vec{B} = \nabla \times \nabla \psi = 0,$$ which means that all magnetic fields are zero, but that can't be!  Can you see where we went wrong? Reading order for Cosmere series by Brandon Sanderson My entry into Brandon Sanderson's works is Mistborn Trilogy, I finished and loved it. Then I got to know that many of his series are connected and happen in the same universe - Cosmere. Next I started with the Way of Kings, but am now kind of worried if I might read the series in the wrong order. So what is the right order to read them? is it the publication order..?   Elantris (2005)  Mistborn Trilogy Warbreaker (2009) The Stormlight Archive: The Way of Kings (2010) Mistborn: The Alloy of Law (2011) The Emperor's Soul (2012, novella) Shadows for Silence in the Forests of Hell (2013, short story published in Dangerous Women) The Stormlight Archive: Words of Radiance (2014)    or I have seen people suggest the following order too,   Mistborn Trilogy Elantris and the Emperor's Soul The Way of Kings Warbreaker Words of Radiance The Alloy of Law Shadows of Self   I have also read posts where they say it doesn't matter which order you read it in. So am confused, since I have already started the Way Of Kings (Only a few chapters in), should I stop it and go back to Elantris, and follow an order, or should I just continue finish this and pick other books in an order. php--glob for searching directories and .jpg only Other than portability, are there any other advantages (or disadvantages) to storing apps on external storage?