Starting from my previous question  and considering that the commutator  $$\left[i\hbar\frac{\partial}{\partial x},x\right]=i\hbar, \tag{1}$$ the associated linear operator momentum (for example the momentum $p_x$ of the $x$-axis) is:  $$p_x\longrightarrow -i\hbar\frac{\partial}{\partial x}=\frac{\hbar}{i}\frac{\partial}{\partial x} \tag{2}$$ The association of $p_x$ with $-i\hbar\partial/\partial x$ is it a postulate or exist a proof that $$\left[i\hbar\frac{\partial}{\partial x},x\right]=i\hbar\color{red}{\boldsymbol{\equiv}[p_x,x]\,\,?} \tag{3}$$
I've been studying the postulates of QM and seeing how to derive important ideas from them. One thing that I haven't been able to derive from them, however, is the identity of the momentum operator.  For simplicity, I'm only thinking about no relativistic effects, no spin, no time-dependent potentials, and one spatial dimension. Also I'm assuming the position operator is simply multiplication by $x$, as in, I'm in position space. So the Hamiltonian operator is  $ H = -\frac{\hbar^2}{2m}\nabla^2+V$.  I know that the momentum operator is $p = -i\hbar \frac{\partial}{\partial x}$.  But how do I get there from the postulates? I know that it makes sense, as it results in the Ehrenfest Theorem, the De Broglie wavelength hypothesis, the Heisenberg Uncertainty Principle (for $x$ and $p$), the momentum operator being the generator of the translation operator, and possibly many other desirable theorems, and correlations with classical momentum.  But none of these are postulates (at least, not in the various formalisms I encountered), so you can't derive $p = -i\hbar \frac{\partial}{\partial x}$ from them. Rather, they are consequences of it. You need to know the operator beforehand to see that they are correct. Yes, this is just semantics, but that is the core issue for me:  Regardless of how much sense it makes, is the identity $p = -i\hbar \frac{\partial}{\partial x}$ (under the assumptions I made) a Postulate, meaning that you can't derive it from other postulates, or can it in fact be obtained from them? And in the latter case, could you show me how?  Note: I know that there are many different and equivalent sets of postulates for QM. But in none that I saw did they name it as a postulate nor properly derived it.
Block network access of a process? I would like to ask if MS Access database supports working in networks &amp; multi-user access? I'm just starting to learn LaTeX. I tried typing \begin{equation*}  (math goes here)  \end{equation*}  and I got a whole bunch of errors. It seems LaTeX interpreted the empty line as the equation: when it saw nothing there, I guess it just ended math mode? It then said my other commands needed $'s so I assume that's what happened. When I typed \begin{equation*} (math goes here) \end{equation*}  everything was fine. So I'm confused. If LaTeX sees an empty line in an environment, does it just interpret this as me changing my mind, and deciding not to use the declared environment? Stupidly, I had been using a condition like this as part of a script:  if [ $(ls FOO* 2&gt; /dev/null) ] # if files named "FOO*" were downloaded  then     echo "Files found"      # ... process and email results else     echo "Not found"     # ... email warning that no files were found (against expectations) fi   That works for zero and one files named FOO*, but fails if there are more than one. From logs I found several different error messages stemming from this:  [: FOO_20131107_082920: unary operator expected [: FOO_20131108_070203: binary operator expected [: too many arguments   My question is: what is the correct way to check, in a Bash if condition, whether one or more files whose name begins with FOO exist?  GNU bash, version 4.2.25(1)-release (x86_64-pc-linux-gnu) How to find $\mathbb{E}(X\mid\mathbf{1}_{X<Y})$ where $X,Y$ are i.i.d exponential variables? Given that $X,Y\stackrel{\text{i.i.d.}}{\sim}\text{Exp}$ with mean $\lambda(&gt;0)$ and $Z=\mathbf{1}_{X&lt;Y}$, what is $\mathbb{E}(X\mid Z=z)$ ?   We have $Z=1$ with probability $\Pr(X&lt;Y)$ and $Z=0$ with probability $1-\Pr(X&lt;Y)$.   But $\Pr(X&lt;Y)=\frac{1}{2}$ as $X$ and $Y$ are i.i.d. random variables.   Does this mean I can simply say that $Z\sim\text{Ber}\left(\frac{1}{2}\right)$ ?   Then, $\mathbb{E}(X)=\mathbb{E}(X\mid Z=0)\times\frac{1}{2}+\mathbb{E}(X \mid Z = 1) \times \frac{1}{2}$  $\implies2\lambda=\mathbb{E}(X\mid Z=0)+\mathbb{E}(X\mid Z=1)$.   But I cannot conclude anything from here. I couldn't find the conditional distribution of $X\mid Z$ either. Maybe I have to condition on another variable $U$ so that $\mathbb{E}(X\mid Z)=\mathbb{E}\,[\mathbb{E}(X\mid Z,U)\mid Z]$. "In" and "on": How can I decide which one to use for vehicles? Examples:   In a car, van, etc. On a bus, boat, motorcycle, etc.   How can one decide which preposition to use? Is memorization the only way or is there a better way?  Note: People generally explain this by either distinguishing between open and closed vehicles or between large and small vehicles. However, the examples I've given defeat both explanations. I have been invited to write a book chapter in an edited volume, relating to my field of work. I have a paper ready that I have not published and was planning to submit it soon. So, I am wondering is it worth to use that article for the book chapter, as both are on the same topic, or is it in a sense a waste of a journal paper, as book chapters might not be seen to have the same academic value? What should I do? Should I, instead, send the ready paper to a journal for a review and write a book chapter, based on a couple papers I have published, for instance?  Generally speaking, where does a book chapter stand? Is it something between a conference paper and a full-fledged journal article? Or, does does it have the same value as either of the above-mentioned publication types, in your opinion? I have recently analysed my website with pagespeed addon on firebug. It suggested me to set expiration on CSS, JS and image files.  I am wondering how do i do this? How to run batch file without admin privillege? Present and future of bots on Stack Exchange There are discussions on Meta already for some time now about the potential usage of bots on Stack Exchange (SE). The earliest I could find goes back to  , but there have been others , .  Here’s an  about having a bot that tries to answer “easy” questions. The scenario is that (almost) duplicate, simple questions are asked time after time, despite SE’s efforts to prevent this by listing related questions at posting time. This is where the bot could be of service: it could quickly retrieve and post an answer from a previous duplicate (or with smarter Artificial Intelligence - AI)  even from elsewhere on the web, like a user manual or a mailing list discussion). I imagine this might save some review time for moderators, and perhaps users would also enjoy not having to answer over and over again to the same question. Besides, answers would be very fast, which should make everyone happy if they are also of good quality.  This idea of “janitor” bots can have more use cases besides answering “easy” questions (see also discussion ), like suggesting missing tags and re-formatting what should have been code blocks, to name a few.  Yet, despite a long time since these discussions, better Machine Learning Classifiers (ML) and AI over the years, and   , it doesn’t seem that any such bots are among us (except maybe ). As part of a research project at the university, I am trying to understand why this is the case.  For this reason I’d like your opinion on the following:   Do you consider a bot “harmful” for the SE communities?   If yes, would you consider the bot harmful even if   it operates in an automatic manner only in limited scenarios (e.g. specific tags) it operates in a semi-automatic manner (namely with the supervision of a moderator who decides for posting or not the answer).  it operates only for a limited amount of time in a small community (e.g. here).   Would you consider it too risky to give a bot a try? Would you accept an answer provided by a bot?   If not, why ? (e.g. you don’t trust replies from machines)  Would you consider useless a “sloppy” answer by a bot, even if it is provided in just a few seconds? Answers may be not perfect, but they may link to other relevant threads that can help the asker to fix the problem without waiting extra time for a human response. How is it like to live in Helsinki?  What is it like to live in Helsinki?  How is living in Helsinki?  What are the differences between these three sentences and what is the common way to ask someone for their opinion about the city they live in? How would you explain covariance to someone who understands only the mean? ...assuming that I'm able to augment their knowledge about variance in an intuitive fashion (  ) or by saying: It's the average distance of the data values from the 'mean' - and since variance is in square units, we take the square root to keep the units same and that is called standard deviation.  Let's assume this much is articulated and (hopefully) understood by the 'receiver'. Now what is covariance and how would one explain it in simple English without the use of any mathematical terms/formulae? (I.e., intuitive explanation. ;)  Please note: I do know the formulae and the math behind the concept. I want to be able to 'explain' the same in an easy to understand fashion, without including the math; i.e., what does 'covariance' even mean? What is the meaning of “runneth” in ? Do I need to take my weight off the bike to lock out the suspension? A story about a guy who helped humanity by being held in a prison I remember a story I have read buy could not remember the author the name or anything apart from the basic plot. The story is about some future where humanity has to fight with some aliens (not 100% sure). The story tells about a guy who was some kind of activist and who was in a prison two or more times and somehow helped humanity by being held in a prison (can't remember details). I think the story starts by saying that there is a statue in his honor that is not very well known but he did more to help humanity than famous politicians.   The story is most likely written by some famous English-speaking author, though I read it in a translation. I'm working on a python script that passes file locations to an scp subprocess. That's all fine, but I'm in a situation where I may end up concatenating a path with a filename such that there's a double '/ in the path. I know that bash doesn't care if you have multiple file separators, but I'm wondering how exactly that is rectified. Is it bash that strips extra /s or does it really not matter ever?  I ask because it will save me several lines of code to check for extra /s while concatenating. I know it's not a big deal, but I'm curious as well. I have a bash script that has the line cd //usr (instead of cd /usr), which seems to imply there might be a significance to using multiple /s in a path Rename command not found when running a script I'm trying to to execute a script, which uses the command rename.   When I execute it, I get a message rename: command not found. Any ideas?   if [[ -z '$3' ]]; then     shopt -s dotglob     rename "$EXPRESSION" ./* else     rename "$EXPRESSION" "$3" fi How does one install PlayOnLinux? How do I install PlayOnLinux on Ubuntu? Gravitational potential at a point in a gravitational field is defined as work done per unit mass in bringing a small test mass from infinity to the point.  I know that:   By integration, we find that gravitation potential is negative. The maximum value of gravitational potential which is at infinity distance is 0, therefore minimum value should be negative.   But, what I do not understand is force is in the same direction as displacement so why work done is not positive? How can I query SEDE for tags I've created? How can I query the  for tags I've created?