Almost all uncertainties (for example the position-momentum uncertainty or time-energy uncertainty) are greater than ${\hbar}/{2} $. But what is the derivation of this  by Heisenberg? Is there any sort of intuitive explanation behind the magnitude of uncertainty? I know why uncertainty happens but I do not why the value. It'd be great if somebody could provide a simple explanation.  $$  \mathrm{uncertainty} \geq  \frac{\hbar}{2} $$
Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?  The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.  This is the mentioned derivation:     $\sin \theta = \frac{\lambda}{\Delta y}$      assuming $\theta$ is small:      $\tan \theta = \frac{\lambda}{\Delta y}$      de Broglie equation:      $\lambda = \frac{h}{p_x}$      $\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$      but also:      $\tan \theta = \frac{\Delta p_y}{p_x}$      equalize $\tan \theta$:      $\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$      $\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$      $\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$      $\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$   So: Which one is correct and why?
I guess I'm asking for comments, with subsequent notifications, to be added to queries in SEDE, so we can point out [fixes]  to flawed queries, especially when [the query]  has a comment in it explaining the problem. Firefox doesn't launch in Ubuntu 13.04 Firefox starts when I use sudo in the terminal, but not from the launcher. All the other apps work like a charm, just not Firefox. How can I fix this? What type of path in shebang is more preferable? The Problem  Accessing my site over https (the only protocol allowed) I get the content, but none of the CSS, images etc.  Drupal is creating pages in response to https requests that include http links to these resources, but it should be using https.  What I've done  I have copied a site to a dev server. The site loads and basically works fine - I can log in etc. It's just that all the links drupal generates are http. The site I've copied it from also runs over https. The site does not exist on http (it would just get a Forbidden error). I've drush cc all.  So Apache's working - I can access the site. .htaccess is working (clean urls work, for example).  I'm using SNI to access the site because the server has other vhosts too, but I'm only accessing the site using the latest Firefox (and as it's a dev site and I'm the only dev, nobody else will use it).  I've done this dozens of times and am quite stumped! Why does Drupal not notice that the request came over https?  The fix  Turns out that .htaccess has this to say:   /* Base URL (optional).                                                          If Drupal is generating incorrect URLs on your site, which could             be in HTML headers (links to CSS and JS files) or visible links on pages     (such as in menus), uncomment the Base URL statement below (remove the       leading hash sign) and fill in the absolute URL to your Drupal installation.    Sure enough, putting $base_url = 'https://example.com'; fixed it.  But why is this necessary? I am a medical doctor in Pakistan. I have been on a career break to allow myself to study full time for the IELTS exam (from which I recently got the required band score) and PLAB 1. I have now registered and paid for the PLAB 1 exam in UK for June 2016, as the next sitting in Pakistan is in November 2016 and I feel like I am ready to do the exam sooner.  Due to having no current employment, my savings are limited (around 25,000 Pakistani Rupees) but I live with my parents and do not have any financial commitments at home.  My savings have depleted in the past ten months whilst I have taken a career break and the money has been used for my own personal expenses. My trip to the UK is being funded by my grandfather who is a British citizen. He is a retired doctor with a good NHS pension. He has provided me with:   6 months bank statements which clearly show his pension deposits   Letter from NHS Pensions stating his pension   Sponsorship letter stating that he will pay for my PLAB exam fee, travel, accommodation and subsistinence costs whilst in the UK   Copy of British passport  Document confirming that he owns the house he lives in mortgage-free Confirmation email from GMC stating he has paid the PLAB exam fee.   Without large personal savings and no current job (but I have job offers now) would my application be rejected or is the strong sponsorship going to be good enough?  Are there any other documents I should consider submitting that may help me get a visa? In the  there are lots of numbers on display, including 1000, 2500 or 10000. The display is erratic though:    Same number, two displays. Both appear valid to me: 1000 or 1,000 but please, make it consistent. My eyes hurt! :) I just found , but there it's discussing the whole network while I'm asking just for that specific page. For a start. I'm connecting to some enterprise wireless network (eduroam).  The thing is that it saves the User/Password in the wireless settings, anyone can click 'Show password' and see my university account password.  What I want is that it prompts me everytime the credentials inputs when I try to connect to the network and don't save it under the 'settings' page.  Is this possible? If yes, how?  Thanks  Ubuntu 18.04 LTS What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime? 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 ? Integral of Wiener Squared process I don't have a background of stochastic calculus.   It is known  that definite integral of standard  from $0$ to $t$ results in another Gaussian process with slice distribution that is normal distributed with mean equal to $0$ and variance $\frac{T^3}{3}$ i-e  $$ \int_0^{t} W_s ds \sim \mathcal{N}(0,\frac{t^3}{3})    $$  Question:  What if we square the standard Wiener process and then integrate i-e $$ \int_0^{t} W_s^2 ds \sim ?    $$  Would that be scaled Chi-square distributed ? How to create a fancy table of contents for Beamer How can I create a fancy-looking table of contents for a Beamer presentation?    \documentclass{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc}  \begin{document}  \frame{ \tableofcontents }  \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{} \section{Foo bar} \frame{}  \end{document} Must present perfect tense be used if the action takes place more than once? I was told that if an action is completed once in the past, the simple past is used. Ex:  I saw that movie.  If the action is completed more than once, the present perfect tense must be used. Ex:  I have seen that movie twelve times.    So, "I have seen that movie twelve times on Tuesday." is correct and "I saw that movie twelve times on Tuesday." is incorrect? While proving a universal quantification, we begin by "Let ... in ..." or "Arbitrary choose ..."? I cannot update my Kali Linux, when trying to execute apt-get update I get this error message:  # apt-get update  Get:1 http://kali.mirror.garr.it/mirrors/kali kali-rolling InRelease [30.5 kB] Err:1 http://kali.mirror.garr.it/mirrors/kali kali-rolling InRelease   The following signatures were invalid: EXPKEYSIG ED444FF07D8D0BF6 Kali Linux Repository &lt;devel@kali.org&gt; Reading package lists... Done W: GPG error: http://kali.mirror.garr.it/mirrors/kali kali-rolling InRelease: The following signatures were invalid: EXPKEYSIG ED444FF07D8D0BF6 Kali Linux Repository &lt;devel@kali.org&gt; E: The repository 'http://kali.mirror.garr.it/mirrors/kali kali-rolling InRelease' is not signed. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   If you need my kernel version:  # uname -a 4.13.0-kali1-amd64 #1 SMP Debian 4.13.10-1kali2 (2017-11-08) x86_64 GNU/Linux   How can I fix this? Why can't the Klein-Gordon equation explain the hydrogen atom but the Dirac equation does? Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom?  Why can the first order Dirac equation explain it? What are the failures? Encode/Decode URLs in C++ Does anyone know of any good C++ code that does this? How is it possible for the Root Name Servers to handle all DNS requests? I was reading about DNS some days ago and learned how the requests are processed. If you surf to www.example.com, then a request will go to the Root Name Servers to see who owns that .com address, then another request will go to another, more local, DNS server to see who owns the example.com address and so on.  How is it technically possible that the 13 Root Name Servers can handle all requests done by earth's billions of Internet users simultaneosly without beeing ddos:ed? dropping trailing '.0' from floats I'm looking for a  way to convert numbers to string format, dropping any redundant '.0'   The input data is a mix of floats and strings. Desired output:  0   --> '0'  0.0 --> '0'  0.1 --> '0.1'  1.0 --> '1'  I've come up with the following generator expression, but I wonder if there's a faster way:  (str(i).rstrip('.0') if i else '0' for i in lst)   The truth check is there to prevent 0 from becoming an empty string.  EDIT: The more or less acceptable solution I have for now is this:  ('%d'%i if i == int(i) else '%s'%i for i in lst)   It just seems strange that there is no elegant way to handle this (fairly straightforward) case in python. Cardinality of the Irrationals Possible Duplicate:          We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real numbers, is uncountable. What can you say, then, about the cardinality of the irrational numbers?  I would say that the cardinality of irrational numbers is uncountable. That would be because the set $\mathbb{Q}$ is a subset of $\mathbb{R}$. The irrationals is also a subset of $\mathbb{R}$. Basically, my thought is that the continuum is composed of irrationals and rationals (only two subsets of reals). Because we can prove that the rationals are countable by Cantor's proof, then the remaining subsets of $\mathbb{R}$, the irrationals, has to be uncountable for $\mathbb{R}$ to be uncountable. My concern is how to go about the prove this (rigorous, intro level). Many years ago I read a book or story where a group of people discover a dark city - maybe on another planet - which appears devoid of life but is in very good condition. As they gradually explore the city one of the them falls into a trap and is cut to pieces by something mechanical. This detail I can remember as it was a pretty gruesome idea. I don't know the ending but my guess is that none of them make it out alive. Expectation of Exit Time of Brownian Motion from Interval I am trying to solve the following:  Let $W_t$ be a brownian motion and $a,b &gt;0.$  Define $\tau$ to be the exit time of $W_t$ from $[-a,b]$, that is $$\tau = \inf\{t \ge 0\ :\ W_{t}(\omega) \notin [-a,b]\}.$$  Show that $\tau$ is integrable and compute $\mathbb{E}(\tau)$.  I have seen this stated without proof multiple places that $\mathbb{E}(\tau)=ab.$  What is the logical argument that backs up this statement?  How are we guaranteed $\tau$ is integrable?