Studying quantum mechanics I have encountered several times in proofs this expression: $$ \frac{\mathrm{d}\langle\phi|\hat{H}|\psi\rangle}{\mathrm{d}t} = \biggl\langle\frac{\mathrm{d}\phi}{\mathrm{d}t}\bigg\rvert\hat{H}\bigg\lvert\psi\biggr\rangle + \biggl\langle\phi\bigg\rvert\frac{\mathrm{d}\hat{H}}{\mathrm{d}t}\bigg\lvert\psi\biggr\rangle  + \biggl\langle\phi\bigg\rvert\hat{H}\bigg\lvert\frac{\mathrm{d}\psi}{\mathrm{d}t}\biggr\rangle  $$  What does the derivative of an operator mean mathematically ? Is something that can be rigorously mathematically defined? Or is just the juxtaposition of the derivative and the operator?  Moreover I was wondering if it is possible to derive this form of Liebniz rule. If I explicitly do the integrals I can carry in the time derivative, but I don't understand how to apply a derivative to a mathematical object which is defined only by its action on a well defined function.
In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.  There are several examples of these. One of them is when we work with the Heisenberg picture, where we need to consider functions $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ such that $\alpha(t)$ is the operator at time $t$.  Another example is when we deal with exponentiation of operators, like when building the time evolution operator:  $$U(t,t_0)=\exp\left(-i\frac{H}{\hbar}(t-t_0)\right),$$  Here, the $\exp$ is usually understood as being defined via the eigenvalues of $H$.  The point is, the idea of a function $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ appears quite often in Quantum Mechanics, and sometimes one needs to differentiate these. In practice we do it formally, using all the properties we would expect, but I'm curious about how one would properly define this.  If we were dealing with bounded operators, then we could use the operator norm, which is available for this kind of operator, and define the derivative as we can usually we do when there's some norm around.  The point is that in Quantum Mechanics most of the time operators are unbounded.  So in the general case, how can one define the derivative of one operator-valued function?
How to play MP3 files using Rhythmbox? Zero search results when searching for [tagOne] [or] [tagTwo] score:-2 I'm able to search for the newest questions tagged with [xcode] and [ios] and score >= -2.    But I see 0 results when searching for the newest questions tagged with [xcode] or [ios] and score >= -2. Ubuntu 14.04 has visual artifacts and freezes I am having difficulty with this proof. I have previously proved that for an automorphism on the extension field $K$ of $F$ that when restricted to $F$ is the identity automorphism must take roots of $f(x)\in F[x]$ in $E$ to roots of $f(x)$ in $E$. I was thinking of using the minimal polynomial $x^{3}-2\in\mathbb{Q}[x]$ and the result of previous problem I solved, but I suspect there are automorphisms that don't fix the subfield, so the result can't be used or I need to prove that any automorphism of an extension field must fix the subfield that is being extended or always sends roots in $E$ to roots in $E$.  I am not asking for a solution to this, but maybe a hint or pointer in the right direction. This problem is introduced in the text prior to Galois theory, so please keep that in consideration. I understand using phone while it is being charged by a normal charger is not advised. What about using phone while it is being charged by a portable charger (powerbank)?  I own a HTC one and i was told by a mobile technician HTC’s own portable charger only should be used for HTC one. How much truth is there to it as HTC powerbank is not available in my city and i am thinking of using Sony portable charger for charging my HTC One? Repeatedly taking differences on a polynomial yields the factorial of its degree? I had recently faced a problem:     Solve the Diophantine Equation $x^2 - y! = 2001$.   Solving it was quite easy. You show how $\forall y \ge 6$, $9|y!$ and since $3$ divides the RHS, it must divide the LHS and if $3|x^2 \implies 9|x^2$ and so the LHS is divisible by $9$ and the RHS is not. Contradiction. Hence, the only solution is $(45, 4)$.  That made me wonder, how we can solve the Diophantine Equation $x^2 - y! = 2016$. We cannot apply the same logic here. $2016$ is a multiple of $9$ and it is clear that $3|x$ and $9 \nmid x$. How do I proceed from here? What set of books makes a complete starter set for 4e? I've asked several questions about 4th Edition D&amp;D because I may be able to start a game. However, I'd need to purchase (some) materials. I am aware of the 4 book set (core) and the expected Essentials Red Box.  So now I need to know: presumably buying either the 4 book set or essentials will be enough to get my game on. Would there be something 'lacking' though? Would I be missing character types, for instance, as I've seen that the splatbook strategy is still in effect?  I'm an experienced player and GM and plenty creative, so many holes I'd be able to fill in myself. I just don't want to get sucked into too much financial impact. How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $? Show that  $$\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i} =0 $$  I've proved that this sequence converges (it is bounded and decreasing). NOW, I need to find a sequence that is bigger than this one and goes to zero. Maybe something using geometric serie of 1/2  Thanks in advance! Sharing Open Source Python/ArcPy Code and Models for ArcGIS Desktop? I will be writing scripts for ArcGIS Desktop in Python with ArcPy.   Is there a community or open source project where code and models can be shared? Why does Microsoft Word 2010 show an error for the following sentence?     1.&nbsp;Where should this car be parked?   Word 2010 also suggests changing the sentence to     2.&nbsp;Where this car should be parked?   Is the sentence (1) wrong? If yes, why? What about sentence (2)? Deploy Community Builder audience page assignments with ant migration tool? In Community Builder, if you create audiences and assign them to page variations, you cannot deploy these using change sets, according to .  In fact, Audience doesn't even appear in the list of change set components.  On the other hand, Audience metadata can be deployed using the ant migration tool, so I was hopeful that the audience assignments would also be deployable.  Unfortunately, I haven't had success in deploying any audience assignments, either for entire pages or for components on a page.  Has anyone found a workaround for deploying audience assignments for Community Builder pages?  I should say that I'm using the Build Your Own template.  If it can't be deployed, is there at least a way to get a list of audience assignments so I don't have to open every single page in the Community Builder to check?  Here's what I'm using in my package.xml manifest.  All the community builder page variations and all the audiences are created correctly, but the assignments are not.  &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;Audience&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;CustomSite&lt;/name&gt; &lt;/types&gt; &lt;types&gt;     &lt;members&gt;*&lt;/members&gt;     &lt;name&gt;SiteDotCom&lt;/name&gt; &lt;/types&gt; No, actually I didn't, but that was a close call and I'm really curious how would one go about that one.  On my third or fourth visit to the Nether, I still didn't realize the risk, and didn't place marks that would help me get back to the origin (currently I like to use flint and iron to create big flaming arrows towards the exit, visible from afar.) I wandered past one hill, and then I couldn't locate the place I came from. I wisened up and started placing marks in form of torches, towers of netherrack with glowstones on top etc. The bunch of wood I had in my stash has proven invaluable as I ran out of steaks, and had to craft a bowl and start making mushroom soup. I realized the fastest way back would be to simply die, but since I had no clue where I was, that would mean losing my quite precious equipment, which cost me a lot of time and effort to gather and enchant, for that trip specifically. I also knew I had traveled rather far from the portal before starting placing marks, so the chances of finding it again by going in any random direction were rather slim.  By sheer luck I found one of the caverns I had explored before, and found my way back, but that was really a close call and I'd prefer to know any hints what to do in Nether if you're already lost (no Obsidian to build a new portal, not willing to die and forgot the direction to the exit.) I'm running an SSH server on my personal computer. The log had many people(or bots?) repeatedly trying to log in to my server (that is before I changed the default port), which made me a little freaked out. I was worried about their dictionary attacks or brute force password guessing.  But then if you type wrong password (or wrong username), the server waits for about 2 seconds before it asks for password again. So an attacker can try one password per 2 seconds. So I concluded that these attacks aren't that great a threat (as long as your password does not consist of dictionary words.) Am I correct? Is there a way to absolutely force a Log Out, Shut Down, or Restart with open tabs and unsaved documents? I could not find any relatively simple way to log out of a public use Mac quickly with many tabs open and unsaved document in the background without resorting to sudo shutdown -h now.  The reason I ask is that sometimes a user will have his/her account logged into the iMacs here at school and they would try to log off and then leave, thinking it has completed but the process is interrupted by Safari and Office apps, leaving their account, mounts, and websites still accessible to the next person who sits down.  Obviously a shortcut string would be nice instead of posting a note telling the person to type arcane things in a terminal. For example, just having them try to find Terminal.app is going to be a chore.  This doesn't seem to make sense because the user can just quit each app and click a few dialog boxes but unlike you and I to whom repeatedly pounding Command-Q while hitting the spacebar and D keys are probably second nature. Google sheets split multi-line cell into new rows (+ duplicate surrounding row entries) Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable What are the differences between executing shell scripts using "source file.sh", "./file.sh", "sh file.sh", ". ./file.sh"? Have a look at the code:  #!/bin/bash read -p "Eneter 1 for UID and 2 for LOGNAME" choice if [ $choice -eq 1 ] then         read -p "Enter UID:  " uid         logname=`cat /etc/passwd | grep $uid | cut -f1 -d:` else         read -p "Enter Logname:  " logname fi not=`ps -au$logname | grep -c bash` echo  "The number of terminals opened by $logname are $not"   This code is used to find out the number of terminals opened by a user on the same PC. Now there are two users logged on, say x and y. I am currently logged in as y and there are 3 terminals open in user x. If I execute this code in y using different ways as mentioned above the results are :  $ ./file.sh The number of terminals opened by x are 3  $ bash file.sh The number of terminals opened by x are 5  $ sh file.sh The number of terminals opened by x are 3  $ source file.sh The number of terminals opened by x are 4  $ . ./file.sh The number of terminals opened by x are 4   Note: I passed 1 and uid 1000 to all these executables.  Now can you please explain the differences among all these? Kill some processes by .exe file name A novel where a woman from "future" of 1999 speaks leet-speak