in WKB the quantization conditions are  $$ \oint _{C} p.dq =2\pi n \hbar \tag{1}$$  and the wave function is $$ \Psi(x)= exp \left( \frac{iS[y(x)]}{\hbar} \right), \tag{2}$$  but what boundary conditions must I impose to this wave function if I want to obtain borh sommerfeld conditions ??
How can one prove the Bohr-Sommerfeld quantization condition $$ \oint p~dq ~=~2\pi n \hbar $$ from the WKB ansatz solution $$\Psi(x)~=~e^{iS(x)/ \hbar}$$ for the Schroedinger equation? With $S$ the action of the particle defined by Hamilton-Jacobi equation $$ \frac{\partial S}{\partial t}+ \frac{(\nabla S)^{2} }{2m}+V(x)~=~0 .$$
Man is sitting in his room and gets contradictory advice from two men from the future Close and flag links do nothing on mobile site What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ?  Thus $\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+...=$ ??  How do I efficiently decide if $\sum_{i=1}^\infty\dfrac{1}{p_{p_i}}\gt1$ is true without using a computer ?  I assume this infinite sum is an irrational number. Has this already been proven ? Nested Quantifiers true or false I have a concern with nested quantifiers.  I have: $$  \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$   My first question, can it be read like this:  $$  \forall x \forall z \exists y(x^2-y+z=0)  $$  The way I did it, is I started off with $x=1, z=1 $   $$ 2-y = 0 $$ $$ y =2 $$  Is this a good approach? Is there a way to find the number of files of a specific type without having to loop through all results inn a Directory.GetFiles() or similar method? I am looking for something like this:  int ComponentCount = MagicFindFileCount(@"c:\windows\system32", "*.dll");   I know that I can make a recursive function to call Directory.GetFiles , but it would be much cleaner if I could do this without all the iterating.  EDIT: If it is not possible to do this without recursing and iterating yourself, what would be the best way to do it? How do you declare an interface in C++? How do I setup a class that represents an interface?  Is this just an abstract base class? Can we have threaded comments? The conversation often continues in the Comments and important information is held in the comments. I'd like to have threaded comments there. Fantasy novel about kid who controls magic through coin changer device I read this book in high school, and I can't remember what it's called. Some kid learns he has magical ability, but only finds out later that he controls it through some kind of coin changer he's been tinkering with the whole book. "An hour" or "a hour" Which indefinite article should precede hour — a or an?        an hour     a hour      Does the usage of an vs a depend on the pronunciation — a history, a hobby, but an hour, an honor? Export My Stack Overflow Data Is there a first party (or third party?) way to export just my data out of Stack Overflow?  I realize there's a massive Creative Commons data dump.  I'm interested in knowing if there's a quick and easy way for me to get my posts dumped out in some plain text file format for easy  personal archiving. GRUB rescue problem after deleting Ubuntu partition! After I deleted my Ubuntu partition, I rebooted my computer and I only get a prompt:   grub rescue &gt;   The command ls lists   (hd0) (hd0,msdos5) (hd0,msdos3) (hd0,msdos2) (hd0,msdos1) (hd1) (hd1,msdos1)   I check them one by one:  ls (hd0,msdos5)/ ls (hd0,msdos1)/   But I cannot find the right partition. Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$. Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this:  You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = \big(\frac{x+1}{x}\big)^x$  And you can express $1-\frac{1}{x}$ as $\frac{x-1}{x}$, such that $\big(1-\frac{1}{x}\big)^x = \big(\frac{x-1}{x}\big)^x =\frac{1}{\big(\frac{x}{x-1}\big)^x}$  Now $\big(\frac{x}{x-1}\big)^x$ and $\big(\frac{x+1}{x}\big)^x$ look very similar, and I can imagine (and see on Mathematica) that $\lim_{x \to \infty} \big(\frac{x}{x-1}\big)^x$ also approaches $e$.  However I'm clueless about limit proofs, how do you prove that?  EDIT: Just had a last second insight.  Can you say that $$\lim_{x \to \infty} \big(\frac{x}{x-1}\big)^x = \lim_{x \to \infty} \big(\frac{x}{x-1}\big)^{x-1} \cdot \lim_{x \to \infty} \big(\frac{x}{x-1}\big)$$ $$= \lim_{x \to \infty} \big(\frac{x}{x-1}\big)^{x-1} \cdot 1$$ $$=\lim_{x \to \infty} \big(\frac{x+1}{x}\big)^x = e ?$$   (not sure about that last limit transition...I mean I'm pretty sure it's true, just not sure how to write it; I would appreciate feedback on that, thank you) Today I updated to version 2.18.52 &amp; I can’t see the names of the people who send me a message. The only thing I see is the WhatsApp logo and the word “message”.    I can’t also see the status my friends posts, they all look black. I uninstalled the app, reset my phone &amp; nothing seems to work so far. I need backup in pressing my case that the phrase “choose from one of four options” is grammatically incorrect. Is there some resource that can prove my case, that the incorrect phrase should be replaced with one of the two following ones?        Choose one of four options   Choose from four options Cannot determine size of graphic I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. I was reading French's Vibrations &amp; Waves where he discusses Huygens-Frensel Principle.   The principle talks about how secondary sources give rise to secondary wavelets to form the displaced wavefront. However, any secondary source can form two wavelets one moving forward &amp; other towards the original source as pointed by French:     [...] The Huygen's construction would define two subsequent wavefronts , not one. In addition to a new wavefront farther away from the source, there would be another one corresponding to a wavefront back toward the surface, But we know this does not happen.    Then he writes:     If the Huygens way of visualizing wave propagation is to be acceptable, it must introduce the unidirectional property of a traveling wave. This can be achieved by requiring that the disturbance  starting out from a given point in the medium at a given instant is not equal in all directions. Specifically, if $O$ is the true original source ,&amp; $S$ is the origin of Huygens wavelet, &amp; $P$ is the point at which the disturbance is being recorded, then the effect at $P$ due to the region near $S$is a function of $f(\theta)$ of the angle $\theta$ between $OS$ &amp; $SP$.     I am not understanding how his reasoning actually averts the possiblity of formation of back-waves. Can anyone help me visualise what he is talking? How does his reasoning maintain the unidirectional propagation of waves? Can anyone explain his argument? Why (and how) did using cat on binary files mess up the terminal? $SL_n(\mathbb R)$ is a subgroup of kernel of $f\in$ Hom$(GL_n(\mathbb R), A)$, $A$ is an abelian group. Try to prove the following proposition.     $SL_n(\mathbb R)$ is a subgroup of kernel of $f\in$ Hom($GL_n(\mathbb R), A)$, where $A$ is an abelian group.   My first attempt is to show that $SL_n(\mathbb R)$ is in form of $ABA^{-1}B^{-1}$, then it's clear that $ABA^{-1}B^{-1}$ is in $ker(f)$. However, there exists an elementary row reduction matrix, i.e. $E_{ij}(m)$ adding m times $j$-th row to $i$-th row. Then $E_{ij}(m)\in SL_n(\mathbb R)$. Now, it is possible that $E_{ij}(m)$ is not in $ker(f)$. Then I try to find any relation between $\mathbb R^{\times}$ and $A$, but, except knowing they are both abelian groups, I cannot figure out what else connects those two group. Could you give me some hints? Change \textwidth after first page in scrlttr2 My primary drive is a 250GB WD SATA drive. So, I added 2x 500GB 7,200 RPM WD SATA drives into my Windows Vista box and created a 1TB RAID0. I then formatted the the primary drive and installed Windows 7. To my pleasant surprise when I booted into Windows 7 my RAID0 was still intact and I kept trotting along the same as I did before.  Now I am replacing my motherboard, processor, and RAM and plan on formatting the primary 250GB drive again and using it to boot for a new clean install of Windows 7.  My question is: if I move these two SATA drives which are setup for RAID0 into the new system, install Windows 7 again, will the RAID0 remain?  Edit: Software RAID. I created it within Windows. The RAID0 does NOT contain the system boot partition.