So I'm reading about bound systems right now in my quantum text. It is beginning to explain why energy must be quantized, and is doing so by introducing the reader to the one dimensional "quanton in a box". Essentially it shows this:  For a region of 0 potential energy, with infinite potential energy everywhere else, a particle is "free" within the region of 0 potential energy. This means that the particles wave function must go to zero at these "boudaries", and therefore resemble a standing wave between the potential energy bounds.  Because the wavefunction must be zero at the ends, the Dr Broglie wavelength must be so that an integer number of half-wavelengths fits inbetween the regions of infinite potential. Therefore, the energy of a particle (quanton) must be so to satisfy this, and cannot be any arbitrary value.  First of all, the more board question: why, then would an electron that is outside of a bound system be quantized? Or any other particle that is outside of a bound system (atomic nucleus, etc.)?   Secondly, the question that I am more interested in: the only reason that a classical standing wave in a string results in the way it does is be cause f resonance; the string resists frequencies that do not match a natural harmonic of the material, and accepts very well frequencies which do match. So then isn't the fact of energy quantization a direct consequence of the phenomenon of resonance? It must be. But then, what is the material that is experiencing this resonance? For an electron, which could be described as a "matter-wave", would the Higgs field, itself, be resonating? Or something else?
Given some potential $V$, we have the eigenvalue problem   $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$   with the boundary condition   $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$   If we wish to seek solutions for   $$ E &lt; \lim_{|x|\rightarrow \infty}V(x), $$   I find textbooks like Sakurai and others assert that (without proof) set of all such eigenvalues $E$ that admits nontivial solutions is discrete, which gives us liberty to index them with natural numbers (principal quantum numbers). Can anyone help me with a little elaboration of this mathematical fact ? As a student of mathematics, I know that compact operators have discrete spectrums, so for the operator $ T : L^2 \rightarrow L^2 $given as  $$ T(\psi) = -\frac{\hbar^2}{2m}\Delta \psi + V\psi. $$   Are there methods to show that $T$ is compact? Or possibly other ways to establish discreteness of its eigenspectrum?
On convergence of Bertrand series $\sum\limits_{n=2}^{\infty} \frac{1}{n^{\alpha}\ln^{\beta}(n)}$ where $\alpha, \beta \in \mathbb{R}$ Study the convergence of $$\sum_{n=2}^{\infty}  \frac{1}{n^{\alpha}\ln^{\beta}(n)}$$ where $\alpha, \beta \in  \mathbb{R}$   I have proved that:   This series diverges when $\alpha \leq 0$. This series converges when $\alpha &gt; 1, \beta &gt; 0$ This series diverges when $0 &lt; \alpha &lt; 1, \beta &gt; 0$ This series converges when $\alpha = 1, \beta &gt; 1$      Question: What happens when $\alpha &gt; 0$ and $ \beta &lt; 0$?   There are other questions on MSE which ask about this series, but this question is distinct because   I would like an argument which does not rely on the integral test for series convergence, and this question considers all real $\alpha$ and $\beta$, while other questions ask only about $\alpha, \beta &gt; 0$, where we can apply Cauchy condensation criterion Determining the name of the VMware host of a VM guest - from the guest Where can I find a large palette / set of contrasting colors for coloring many datasets on a plot? I need large set (20 and more) of contrasted and easily visually-distinguishable colors for coloring different datasets on a scientific plot (the set of colors should not include white or black, and colors should be as bright as possible).   The plot is generated programmatically and intended to be included in a presentation which will be displayed using a projector.   Are there any ready-made sets of colors for such a task (I need colors as a set of RGB values)?   Or maybe there is some algorithm which generates such a set of colors programmatically (I use  for plot generation)?  P.S. The plot itself may look, for example, similar to this one: Does an adjective or longer modifier apply to both nouns when they are joined with 'and' [or 'or']? Can you grab the blue shirts and socks?   Is the above sentence stating that both the shirts and the socks are blue? Or only the shirts?  At this stage, I am leaning towards the earlier (only the shirts) — though writing "Can you grab the blue shirts and blue socks?" seems redundant. $\lim_{n→∞} f(n+x)=0$ results in $ \lim_{x→∞} f(x)=0$ $f\colon [0,\infty) \to \mathbb{R}$ is uniformly continuous.  If $\lim_{n\to \infty}f(n+x)=0$ where $x\in [0,1]$, then $\lim_{x\to\infty} f(x)=0$.  every my trying to solve this is wrong :(  how can i prove this?  and if $f$ is just continuous, what happens to the results How do you prevent web pages from appearing in the Google search index? I have some web pages that Google has indexed. I have now added a robots.txt file to exclude them, but should I also add a noindex tag? If so, then how do I do so? What are the strategies for getting feedback on articles? As an academic, one has to publish. Often, an article is drafted, and you need feedback of a colleague.  I see that quite often some perspective articles are often written by long-term collaborating pairs of authors.  What are some strategies people use to get feedback on their articles  (outside of immediate boss; and in the case that none of the friends in the field of work in the specific domain of the article, and one needs specific (not general) feedback)?   How do you approach a colleague to simply read your article. Or do you just mention it at a conference to the most suitable colleague - would you like to read an article and give me feedback? How do handle the co-authorship or acknowledgement? Do you establish the limits at the "approach time"?  What are some strategies to establish a "publishing" duos (buddies). How do you overlay an image on a shader in Cycles? Difference between "theoretical computer science" and (beta) "computer science" Possible Duplicate:          Title says it all. From what I can see, "theoretical computer science" questions seem to be slightly more isolated to discussing algorithms, whereas "computer science" seems to have questions that are slightly broader and higher-level. Is there any place on the Stack Exchange sites where they give a longer description?  I'm just curious as to what niche the beta "computer science" site is meant to fill, as I see many topics in both forums that overlap. Block IPs if they access a resource Prove that $i^i$ is a real number According to WolframAlpha,  but I don't know how I can prove it. java.lang.StringIndexOutOfBoundsException: String index out of range Now that all SE sites have their own meta site, why doesn't Stack Apps have a meta?  Sure, Stack Apps is somewhat of a limited site (or rather, a site about a limited subject), so by some counts, it might not need a meta. However, I think there are some things that could be discussed about how Stack Apps should be run and there is no good place to do that now. Thus, I support the creation of a Stack Apps meta. When to use 'by' instead of 'at' or 'on' to specify time/date? GPS PPS signal corresponds to data time? I currently have an Adafruit Ultimate GPS connected to an Arduino that outputing GPS data (nmea sentences) once per second.  There is also a pin on the GPS outputting 1 PPS signal.    My question is whether the pulse of the PPS suppose to match the time that the GPS data is coming in i.e are they are coming in at the same time?    Also does PPS pulse come in at the beginning of a UTC second or is it random and simply once a second? How did this question get a 'linked' question that isn't mentioned in comments or answers? Looking at  which has been closed as a duplicate (quite right), we see three 'linked' questions in the sidebar; however, only two of those questions are given as duplicate-close-targets.  which to my mind is the actual 'exact duplicate', isn't mentioned anywhere else except in the 'linked questions' sidebar itself! The  says:  The Linked sidebar... gathers up any links to this question provided by the community via comments, answers, or questions  So how did  get into the linked questions sidebar of ? (There's no explicit link in the other direction, in case you were wondering.) How to add existing user to an existing group? I want to add the Apache user (www-data) to the audio group. I've read the man page for useradd, but I'm not having any luck. I'm running xubuntu 11.10. Here's what I'm doing:  $ sudo useradd -G audio www-data useradd: user 'www-data' already exists   If I leave out the -G option, bash, prints the help info for useradd:  $ sudo useradd  audio www-data Usage: useradd [options] LOGIN Options: -b, --base-dir BASE_DIR       base directory for the home directory...   It's not clear to me from the man page what options I should use to make this work. $y'=$ ${-x+\sqrt{(x^2+4y)}}\over 2$ , $y(2)=-1$ Prove that the solutions $y_1$ and $y_2$ for the initial value problem   $y'=$${-x+\sqrt{(x^2+4y)}}\over 2$ , $y(2)=-1$  are:  $y_1=1-x$  $y_2={{-x^2}\over 4}$  And explain why the two solutions not contradiction with the theory of existence and uniqueness.  ( sorry I don't speak English well )  My answer:  For the initial condition :  $y_1=1-x$ Satisfy the initial condition  $y(2)=-1$  $y_1=1-2=-1$   And Satisfy the differential equation  ${y_1}^{\prime}=-1$  ${{-x+\sqrt{x^2+4y}}\over 2}={{-x+\sqrt{x^2+4(1-x)}}\over 2}$  $={{-x+\sqrt{(x-2)^2}}\over 2}$  $={{-x+x-2}\over 2}=-1$  L.h.s.=R.h.s.  $y^{\prime}={{-x+\sqrt{x^2+4y}}\over 2}$  Also $y_2$ Satisfy the initial condition  $y_2={{-x^2}\over4}={-4\over4}=-1$   And satisfy the differential equation.   $y_2^{\prime}={-x\over 2}$  ${{-x+\sqrt{x^2+4y}}\over 2}={{-x+\sqrt{x^2+4({-x^2\over 4})}}\over 2}={-x\over 2}$  And for second part of exercise:  ${\partial f\over \partial y}={1\over \sqrt{x^2+4y}}$  and this not defined when $x=2, \ y=-1$  So $f(x,y)$ no satisfy lipshtize condition in any rectangular has (2,-1), Lipshtize condition including uniqueness of solution, so no contradict with the existence and uniqueness.  True ?  If this also wrong I well be delet it , I answered by the same way of my teacher :( remove files when name does NOT contain some words How can I restart a Windows 7 computer without screen or mouse? I've recently had problems with my work computer not recognizing either of its attached screens.  I've also had problems with my home computer "locking" the mouse to one app, and no longer acknowledging clicks on, say, the start menu.  In both cases the computer is still running, and I can close apps and so forth with the keyboard.  Restarting usually addresses the problem.  What's the simplest way to restart a Windows 7 computer by the keyboard alone?  Something consistent across all versions would be super.  In Win XP, I believe it was Ctrl-Alt-Delete; pause; Alt-R for Restart; Enter to select.  But on 7 Ctrl-Alt-Delete no longer brings up the same menu.  I'd use Ctrl-Alt-Esc for the Task Manager, but it no longer has a Restart menu (at least on Win 7 Home).