Is it in general? $[\Lambda,\Omega]=i\hbar\{\lambda,\omega\}_{x,p}$ In R. Shankar's book, He has written $$[X_i,P_j]=i\hbar\{x_i,p_j\}=i\hbar$$ Is there any specific reason to use the Poisson bracket? Is there any general relation which looks like? $$[\Lambda,\Omega]=i\hbar \{\lambda,\omega\}_{x,p}$$
What is the connection between Poisson brackets and commutators? The Poisson bracket is defined as:  $$\{f,g\} ~:=~ \sum_{i=1}^{N} \left[  \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} - \frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}} \right]. $$  The anticommutator is defined as:  $$ \{a,b\} ~:=~ ab + ba. $$  The commutator is defined as:  $$ [a,b] ~:=~ ab - ba. $$  What are the connections between all of them?  Edit: Does the Poisson bracket define some uncertainty principle as well?
Electromagnetic Tensor in Cylindrical Coordinates I understand that the  is given by  $$F^{\mu\nu}\mapsto\begin{pmatrix}0 &amp; -E_{x} &amp; -E_{y} &amp; -E_{z}\\ E_{x} &amp; 0 &amp; -B_{z} &amp; B_{y}\\ E_{y} &amp; B_{z} &amp; 0 &amp; -B_{x}\\ E_{z} &amp; -B_{y} &amp; B_{x} &amp; 0 \end{pmatrix}$$  where $\mu$, $\nu$ can take the values {0,1,2,3} or {$t,x,y,z$}.  So, for example  $$F^{01}=F^{tx}=-E_x$$  My question is, what would the following expression be?  $$F^{t\rho}=?$$ or $$F^{z\rho}=?$$  where $\rho=\sqrt{x^{2}+y^{2}}$ is the radial coordinate in ?  And more generally, how can we construct the Electromagnetic Tensor in cylindrical coordinates? Where $\mu$, $\nu$ now take the values {$t,\rho,\varphi,z$}. How do I opt-out of the Social Security system? I'm trying to do a volumetric light on Cycles, and I've followed tutorials, but nothing has worked yet. How can I make a proper volumetric light in Blender, like the following one?    But to leave the light in the middle, like, almost a star, is this possible? I'm at the point to use an image plane besides the light which would be better... I need to use it moving the camera.  I'm actually trying to do this because I want to simulate the brightness of the Sun.  I tried the effect, but it still gave me an error: What could happen if I use 2 cycle fuel in a 4 cycle engine? First off, this is NOT to be confused with the following question:    2 cycle oil is used for my weed eater. There is no way I could use a whole gallon in a year.  Sometimes I run out of gasoline for my 4 cycle lawn mower, and that 2 cycle mix fuel is sitting there getting really old.  If I run that in the lawn mower, am I simply going to create extra pollution or could the 2 cycle fuel cause any damage to the 4 cycle engine? What is the Python equivalent of static variables inside a function? How to read a csv file and use the values as x and y points in blender? How can i use the value in column 'lon' and 'lat' as the position of my uv_sphere? For example,read the values from line 4, (116.4559081, 39.9139083), and make these 2 numbers as my uv_sphere x and y position in blender? and gives different colors for different mode, for example i have 4 modes in the csv , i want to make the uv sphere that represents the bus mode to be red, and subway to be green , rail to be blue and tram to be yellow., and use the row ['from'] and ['to'] to connect the nodes with a cylinder with x scale 0.5 , y scale 0.5 and the z scale with the value in ['length'] value. Do I need to have antivirus software installed? Solve the following equation: $\sqrt {\sin x - \sqrt {\cos x + \sin x} } = \cos x$ Solve the following equation:  \begin{array}{l}{\sqrt{\sin x-\sqrt{\cos x+\sin x}}=\cos x} \\ \text{my try as follows:}\\{\sin x-\sqrt{\cos x+\sin x}=\cos ^{2} x} \\ {\sin x-\cos ^{2} x=\sqrt{\cos x+\sin x}} \\ {\sin ^{2} x+\cos ^{4} x-2 \sin x \cos ^{2} x=\cos x+\sin x} \\ {\sin ^{2} x+\cos ^{4} x-2 \sin x \cos ^{2} x-\cos x-\sin x=0} \\ {\sin ^{2} x+\cos ^{4} x-2 \sin x\left(1-\sin ^{2} x\right)-\cos x-\sin x=0} \\{\sin ^2}x + {\left( {1 - {{\sin }^2}x} \right)^2} - 2\sin x\left( {1 - {{\sin }^2}x} \right) - \cos x - \sin x = 0\\ {\sin ^{2} x+\sin ^{4} x-2 \sin ^{2} x+1-2 \sin x+2 \sin ^{3} x-\cos x-\sin x=0} \\ {\sin ^{4} x+2 \sin ^{3} x-\sin ^{2} x-3\sin x-\cos x+1=0}\end{array}  Now i think it gets more complicated , any help would be appreciated A white number inside a black circle. How to make a white number inside a black circle (the picture below)? I want to change numbers inside the black circle, of course.  I use Share LaTeX. I have read some articles here but anyone are not what I would like to get. I would like to get this  (the picture below) - the sixth answer.   Thanks for help. When writing a book and referring to a female and male aged 18–25, what term would be most appropriate? Boy and girl seem too young, whereas man and woman seem too old.  Example sentence:     A boy/man emerged from the shadows.   Edit: I think the word I am looking for is similar to 'guy' but less American and colloquial. Any thoughts? Solution check: Let $f_n$ be fibonacci numbers. Prove: $\sum_{k=0}^{n-1} \binom{n+k}{2k+1} = f_{2n-1}$ and $\sum_{k=0}^n \binom{n+k}{2k} = f_{2n}$ avoid space after commas used as thousands separator in math mode How to install H.265 / HEVC codec on Ubuntu Linux? I have seen that the new  shows off H.265/HEVC playback, the successor of H.264. I wonder how I may get this support in Linux. Is there a need to install a codec specific for this format? How? Does Time Machine backup programs like Photoshop and iWork? I was trying to prove this question.   Q. If G is a group of even order, prove that it has an element a, such that a$\neq$e satisfying $a^2$=e.   I tried solving it using Principle of mathematical induction.  So, Let O(G)=2m, where m$\in$ $N$.   We proceed by induction on m.   For m=1, O(G)=2, $\therefore$ G contains two elements.  G=$\{{e,a}\}$. Now, e has order 1. So, a can not have order 1. $\therefore$ a has order 2. So, it true for m=1.  Now, let it be true for m=k.  Then, we have a group G of order 2k, where $\exists$ an element a$\in$ G, whose order is 2.   We wish to prove it for m=k+1.   Then G has order 2k+2.   I don't know how to proceed further. Let $f$ be an analytic function such that $\operatorname{Im}(f(z))=\operatorname{Re}(f(z))^2$. Prove that $f$ is constant. How to solve this question? Also it would be great if someone could tell me from which book this question is from? Roots of a functionwith condition $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Let $f:[0,\pi] \rightarrow \Bbb R$ be a continuous function which satisfies $\int_0^\pi f(x) \sin x dx = \int_0^\pi f(x) \cos x dx =0.$ Show that $f$ has at least two roots. Ambiguity of "quite" The adverb "quite" has the following meanings :     1: wholly, completely ("not quite   finished")      2: to an extreme : positively "quite   sure" —often used as an intensifier   with a "quite a swell guy" "quite a   beauty"      3: to a considerable extent : rather   ("quite near")   My understanding is that #1 and #2 conflict with #3. For example, when you say "She's quite right", do you mean "She's completely right" or "To a considerable extent, she's right"? Notice that in the latter case you're not saying that she is completely right.  Another example is "I'm quite sure". Does it necessarily mean "I'm completely sure"?   Does the word really have conflicting meanings that can possibly cause ambiguity? Or am I missing something? How can I get the page numbers only of a pattern in a pdf file, regardless if the pattern is multiline? If a>0 and b>0,  $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $  What I was trying to do: Suppose a>b. Then, for sufficiently large values of x, $ a^x &gt;&gt; b^x $; so $\sqrt[x]{a^x+b^x} \rightarrow \sqrt[x]{a^x} \rightarrow a $ when $x \rightarrow +\infty$.   Is that idea correct? How can I formalize it?