Appendix header and references in sections
References in Chapter Titles
This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching.  The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object.  This naturally leaves students wondering     what is, exactly, the physical significance of the object $[A,B]$ itself?   and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources.    One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   Commutation relations are usually expressed in the form $[A,B]=0$ even though, a priori, there appears to be little motivation for the introduction of such terminology. A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means.  (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the  but that's quite a bit to expect a student to grasp at a first go.) From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.) Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations.  This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special. An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion.   I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important? Expected number of games played in a game with 'win by two' rule. Calvin and Hobbes play a match consisting of a series of games, where Calvin has probability p of winning each game (independently). They play with a “win by two”” rule: the first player to win two games more than his opponent wins the match. What is the expected number of games played? My attempt: I use Markov chains. Let $S$ denote that starting state, $W$ and $L$ denote the states where Calvin and Hobbes win respectively. Then, I have $$\mu_S = 1+ p\mu_W+q\mu_L $$ $$\mu_W = 1+ q\mu_S $$ $$\mu_L = 1+ p\mu_S $$ where $\mu_j$ denote the expected number of games from state $j$. Solving the system, I get the solution $\mu_S = \frac{2}{1-2pq}$ which I think is correct. I however would like to know how to approach it with geometric distribution instead of Markov chains. I read one solution where the number of games is given as $1+\text{Geom}(p^2+q^2)$ but I do not understand it. My understanding is that for the game to stop, the required probability is $p^2 + q^2$ since this is the probability that either Calvin or Hobbes win two games in a row. How to use this information in a Geometric random variable? Thanks. How do you get the current time in milliseconds (long), in VB.Net? In calculating work done by a constant force over a constant distance, why doesn't the subject's initial velocity matter? Assume a point-mass $m$ is travelling in a straight line, and a force $F$ will act on $m$ (in the same direction as $m$'s velocity) over a constant distance $d$; why doesn't $m$'s velocity matter to the calculation of work done on $m$ by $F$? Work is defined such that, in this example, the work done by $F$ on $m$ is equal to $Fd$, but it seems that if $m$ were moving slower, it would spend more time in the field, allowing $F$ more time to act on $m$, thereby doing more work. In fact, if $m$'s velocity were very great, it would hardly spend time in $F$'s field at all (so very little work done). Maybe I misunderstand work; can someone address this confusion of mine? First, when I never had installed sl:  diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   traslaction: "The program 'sl' is not currently installed. You can install it by typing:"  But when I'm tired of it and I remove it by typing sudo apt-get remove sl per example...  I get this when I type sl:  -bash: /usr/games/sl: Ficheiro ou directoria inexistente   translate: "-bash: /usr/games/sl: No such file or directory" instead of:    diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   This appears with others packages too...  How I get rid of this: -bash: /usr/games/sl: Ficheiro ou directoria inexistente ?  I want see the original:  diogosaraiva@Ubuntu:~$ sl O programa 'sl' não está instalado actualmente. Pode instalá-lo escrevendo: sudo apt-get install sl   Output of type sl  sl está hasheado (/usr/games/sl) How to override parent functions in child themes? I've been reading around and trying to figure out how to do this, but for some reason I can't seem to override parent functions in my child theme.  I'm using TwentyTen as a parent - can anyone tell me why this function in my child theme isn't overriding the parent function please?  // Override read more link function osu_twentyten_continue_reading_link() {  return ' &lt;a href="'. get_permalink() . '"&gt;' . __( 'Read on &lt;span class="meta-nav"&gt;&amp;rarr;&lt;/span&gt;', 'twentyten-child' ) . '&lt;/a&gt;'; } function osu_twentyten_auto_excerpt_more( $more ) {  return ' &amp;hellip;' . osu_twentyten_continue_reading_link(); } remove_filter( 'excerpt_more', 'twentyten_auto_excerpt_more' ); add_filter( 'excerpt_more', 'osu_twentyten_auto_excerpt_more' );   I thought you had to remove the filter/action etc. before re-adding it right?  Thanks,  osu Linear regression with fixed point I have a data set to which I want to add a trend line. For my system, there will always be a fixed point on (2, 0). I have the line representing this system, but currently I do not account for the fixed point. Is there functionality to achieve this?  \begin{figure} \begin{tikzpicture} \begin{axis} [     ylabel={Some Time [ms]},     ]     \addplot[only marks, color=red, error bars/.cd,                        y dir=both, y explicit]         table [x=num_of_nodes, y=MyData, y error=stdev_delay_var] {MyData.dat};     \label{plot_one}     \addplot +[mark=none, dashed]          table[y={create col/linear regression={y=MyData}}]         {MyData.dat}; \end{axis} \end{tikzpicture} \caption{Test Results} \label{fig:test_results} \end{figure} How do I remove "Ubuntu" in the bios boot menu? (UEFI) Sort by author in Alpha bibliography Root cause for "Entity is not org-accessible" for sharing object? How to install HPLIP on my Ubuntu to support my HP printer and/or scanner? I've had my Trek 3900 for about a year now, and I replaced the chain about 3 months ago, but otherwise it's running with the same parts it came with (not counting tubes and tires).  I had to replace my chain because it had gotten really rusty from riding in rain/mud/water on trails, and had started skipping.    My problem is, whenever I ride, if I get above or below a very light pressure when pedaling, the chain skips. I was wondering if this could be caused by my chain, which has gotten a little rusty but not as stretched as the other one was, or if I might need a new cassette?    I just readjusted my derailer, but it still skips.   Does anybody know what could be causing this? Why should I put a ~ before \ref or \cite? My question is in the title of this post, but extends to other referencing commands like \eqref, \citep, \citet (and the like) as well. What happens if my shield breaks? If $A \in M_{n}(\Bbb{R})$ is an Upper triangular matrix with diagonal entries $1$ such that  $A \neq I$, then what can we say about the diagonalizability of $A$ ?  I know that if the matrix has distinct eigenvalues or the set of eigenvectors are linearly independent then the matrix is diagonalizable.  And that the eigenvalues of the triangular matrices are given by the diagonal elements like  and , but they work nocely if we had distinct elements on the main diagonal.  But in my case I have same value 1 on the main diagonal, how can I approach about the diagonalizability of the matrix? I'm trying to prove that $e^{\sqrt 2}$ is irrational. My approach: $$ e^{\sqrt 2}+e^{-\sqrt 2}=2\sum_{k=0}^{\infty}\frac{2^k}{(2k)!}=:2s $$ Define $s_n:=\sum_{k=0}^{n}\frac{2^k}{(2k)!}$, then: $$ s-s_n=\sum_{k=n+1}^{\infty}\frac{2^k}{(2k)!}=\frac{2^{n+1}}{(2n+2)!}\sum_{k=0}^{\infty}\frac{2^k}{\prod_{k=1}^{2k}(2n+2+k)}\\&lt;\frac{2^{n+1}}{(2n+2)!}\sum_{k=0}^{\infty}\frac{2^k}{(2n+3)^{2k}}=\frac{2^{n+1}}{(2n+2)!}\frac{(2n+3)^2}{(2n+3)^2-2} $$ Now assume $s=\frac{p}{q}$ for $p,q\in\mathbb{N}$. This implies: $$ 0&lt;\frac{p}{q}-s_n&lt;\frac{2^{n+1}}{(2n+2)!}\frac{(2n+3)^2}{(2n+3)^2-2}\iff\\ 0&lt;p\frac{(2n)!}{2^n}-qs_n\frac{(2n)!}{2^n}&lt;\frac{2}{(2n+1)(2n+2)}\frac{(2n+3)^2}{(2n+3)^2-2} $$ But $\left(p\frac{(2n)!}{2^n}-qs_n\frac{(2n)!}{2^n}\right)\in\mathbb{N}$ which is a contradiction for large $n$. Thus $s$ is irrational. Can we somehow use this to prove $e^\sqrt{2}$ is irrational? How to parse hundred html source code files in shell? I would like to pause a render that I have started, with the intention of resuming it at a later time (but during the same Blender session).  How can I do this? I would like to see a full how-to guide on how to install Ubuntu. So, I have proven already that every open set in $\mathbb{R}$ can be written as a countable union of disjoint open intervals; i.e., that $\mathcal{O} = \bigcup_{i=1}^\infty (a_{x_i}, b_{x_i})$.  Now, consider the closed interval $\left [ a + \frac{1}{n}, b - \frac{1}{n}\right ] \subset (a, b)$.  Since the midpoint of $(a,b)$ is $\frac{b-a}{2}$, we need $\displaystyle n &gt; \frac{2}{b-a}$.  Otherwise, $\displaystyle a + \frac{b-a}{2} = a + \frac{1}{ \frac{2}{b-a}} = b - \frac{1}{ \frac{2}{b-a}} = b - \frac{b-a}{2}$, and our interval will be degenerate.  Note that $\forall (a,b) \in \mathcal{O}$,   $\displaystyle \bigcup_{n = \frac{b-a}{2}+1}^{\infty} \left[ a + \frac{1}{n}, b - \frac{1}{n} \right] = (a,b)$.  So, we have $\displaystyle \mathcal{O} = \bigcup_{i=1}^\infty \left ( \bigcup_{n = \frac{b-a}{2} + 1}^\infty \left[ a_{x_i}+\frac{1}{n}, b_{x_i}-\frac{1}{n} \right] \right)$.  But, I don't like this. It looks like I'm still just taking a union of open sets.  Please help me get this to look like the union of closed sets.  (For unbounded intervals, I know what to do - it should follow from this relatively easily).