In quantum mechanics, if two observables commute, then perfect knowledge can be gained about both observables simultaneously. But what does the commutator actually, physically represent? Like observables correspond to things that can be observed, is there a similar physical meaning or classical analogy for commutators?
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?
Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$. 16.04 VirtualBox VM (from vhd file) hangs at "non blocking pool is initialized" I'm trying to boot Canonical's Azure 16.04 .vhd file in a VirtualBox VM and it hangs at "non blocking pool is initialized".  It works in Hyper-V, so it may only be a bug in virtual box (5.0.20). On 23rd of June, 2016, the UK . This leads to the following questions:   Are UK citizens traveling to the EU affected? Are EU citizens traveling to the UK affected? Are UK/EU family members traveling both way affected? Are Commonwealth citizens traveling to the UK affected? Are other visa-free nationals (such as US/Canada citizens) affected? What other consequences might 'Brexit' have on travelers?   Also, see the  on Expats.SE. How to avoid nested With[]? I am currently looking at the various libraries that I can use to show a map in my custom Android App. I have found several like Mapsforge, OSMDroid, ArcGIS runtime for Android, Google Maps for Android and so on. I'm sure that I am missing a few.   Is there a comparison chart somewhere about the differences in functionalities and pros and cons of each one?   If there isn't one, we could make this question community wiki, to serve alongside  &amp; . Google translate exclude words When using the \cite command in figure captions and the ieeetr bibliography style, then  BibTeX numbers the citations as appearing earlier than they do in reality (i.e. the first citation in the document ends up being [42]).  Is there any way to get BibTeX to ignore the \listoffigures section when numbering? Is there any reference where Captain America's shield is made from stolen Wakanda vibranium? Parallel option performing worse on cluster I'm trying out oracles parallel option on a cluster and surprisingly I am getting worse results with the parallel option. I was expecting some improvement with the parallel option but certainly not worse results. I am wondering why this is the case, and if there is anything wrong with the way I am using the parallel option on my cluster.   I am using a degree of 4 when the number of CPU's I have is 8. I have tried adding parallel to the cluster directly ALTER CLUSTER cluster PARALLEL 4 as well as at the statement on both the index /*+ PARALLEL_INDEX(clust_index, 4) */and tables /*+ PARALLEL(4) */,   Here is my trace from the parallel:    Without parallel: Opening a context menu (with the Menu key) Question Regarding Cardano's Formula In Cardano's derivation of a root of the cubic polynomial $f(X)=X^3+bX+c$ he splits the variable $X$ into two variables $u$ and $v$ together with the relationship that $u+v=X$. From this he finds that $x=u+v$, where  $$u=\sqrt[3]{\frac{-c}{2}+\sqrt{\frac{c^2}{4}+\frac{b^3}{27}}}$$ and $$v=\sqrt[3]{\frac{-c}{2}-\sqrt{\frac{c^2}{4}+\frac{b^3}{27}}}$$  is a root of $f(X)$.      Is there an intuitive explanation for why Cardano splits the variable $X$ into two parts, $u$ and $v$? What's the proof that the Euler totient function is multiplicative? That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by. "Use Option as Meta key" with ability to type the # symbol I am using Terminal version 2.5, where I have two requirements:  I need to be able to type the Meta key. I need to be able to type the # character, too (Option+3 for the UK keyboard layout)  How can I satisfy both of these? By default, (2) is satisfied, but (1) is not. I can then turn on &quot;Use Option as Meta key&quot;, which satisfies (1), but breaks requirement (2). How can I have both? (It shouldn't be so hard for me to just be able to type.) Edit: what I'm looking for, for example, might be a way to just &quot;Use Option as Meta key&quot; for the left/right Option key, or a way to override the &quot;Use Option as Meta key&quot; for the sole exception of the # character. If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$? Keyboard Glyphs How are people showing keyboard &quot;glyphs&quot; on Super User? I looked at the Markdown help page and it didn't show anything regarding this. Here is  (question has been deleted since this was asked, so an archive link is provided instead): Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. T-SQL: How to update just date part of datetime field? How to move boot and root partitions to another drive Order of the Rubik's cube group How to prove the divisibility rule for $3\, $ [casting out threes] The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three.  This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:  $$1212582439 \rightarrow 37 \rightarrow 10\rightarrow 1 \implies 3\not\mid 1212582439$$ $$124524 \rightarrow 18 \rightarrow 9 \implies 3\mid 124524$$  This works for as many numbers as I've tried.  However, I'm not sure how this may be proven.  Thus, my question is:     Given a positive integer $n$ and that $3\mid\text{(the sum of the digits of $n$)}$, how may we prove that $3\mid n$?