Without going deep into mathematics and simply using symmetry arguments I made the following observations-  An electron has a probability of being at a particular position Let's take the 1s orbital of a Hydrogen like single electron species and a uniform sphere at some distance $r$ from it. By Symmetry we can conclude the electron is equally likely to be present at any point on the sphere. Hence for any point P on the sphere there will be a diametrically opposite point Q passing through the nucleus where the electron is equally likely present. Let's say at 2 different times the electron is actually present there so for those instants the mass of electron is present at that point. All of this happens very quickly in fractions of seconds it can be on some point and then at another so for an observer it would seem it's present at multiple points right? If the observer then tries to calculate the average position of the electron taking symmetry arguments wouldn't it lie on the nucleus itself as the average of two diametrically opposite points is the midpoint of the diameter i.e. the nucleus.  With this isn't this somewhat similar to saying the Centre of Mass of the electron lies at the nucleus as the centre of mass is the point where we can assume the mass to be concentrated and we can take similar symmetry methods to calculate say the centre of mass of a uniform solid sphere to be at the centre. This is just something I thought of a while ago, does this make sense to conclude? Also if I were to extend this to multi electron species how would the reasoning go? How would I incorporate the effects of other electronic interactions? Edit: As pointed out by  in the comments the assumption that the electron is at a particular position is wrong, however even if we leave out that part i still feel the symmetry argument should hold
I read Griffiths EM today and it says something very interesting but a little bothering to me. It states for an atom, the position of center of mass of an electron cloud lies in the center of the proton.  But my question is why? What will happen if this is not true?
Can less be used without any comparison? Can I use &quot;less&quot; in sentences like this:  1 Why do we have so less number of students for this class? 2 My song collection is very less.  &quot;Small&quot; sounds better in both examples, but I would like to know if using &quot;less&quot; in these sentences is correct? How is (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ (p ∧ ¬r) ∨ (¬q ∧ q)? Is it really distributive property? I am trying to make an explosion in Blender.  I am having a problem with the smoke simulation with the Cycles rendering engine. More specifically, I have been able to render smoke, but flames do not show up. In the viewport however, everything looks perfectly fine.  The render looks like this:    But in the viewport, it is clear that there are flames. Does anyone have any idea what is going on?     This is NOT a duplicate of  because I have Flame Rate set to 1.0 app crash in the dealloc Arrays in Java and how they are stored in memory Determine number of pages in a PDF file Expression for a choice which isn't really one Toggle Switch to swap the connections of two buttons How can I use only the external flash on slave mode without firing the built-in flash? I have a Canon 700D and YN-568EX ii external flash that I am using in slave mode. But whenever I try to fire the built-in flash pops up automatically and fires itself. Is there any option to use only external flash? How to draw a return arrow from node-3 to node-1 If "clearly evident" is redundant, what word or phrase suggests being indisputable but not readily apparent? A  asks whether the modifier "clearly" changes the meaning of "evident". I would have said, prior to examining a few dictionaries, that evident did not necessarily mean readily apparent, but to my surprise it does.  This brings me to ask what word or expression would mean something can be taken as a true fact, but not through an easily constructed (not obvious or readily apparent) rationale?   For example, if Sherlock Holmes says something is true while everyone else is still sitting on their hands, then it is evident to Holmes, but not to the rest of the world. So, from everyone else's point of view, it is not readily apparent, and (by definition) apparently it is not evident, clear, or clearly evident at all, based on the dictionary definitions. But (let's stipulate) it is true and deducible from the observable evidence. What would we call this?  Update  An update is highlighted above. I can see from some of the answers and comments that I may have gone off my mark with the phrase "not through an easy-to-follow rationale”, and I have modified the question to correct this. I understand that there can be rationales, arguments, or proofs that, once explained, may or may not be easy to follow. What I intend is that the rationale to support the indisputable conclusion is not (initially) easy to recognize or construct (not readily apparent) in the first place. I apologize for the confusion. (The original can be seen in the change history.)  Update 2  Shortly after offering a bounty to this request, I recalled a single word that I now believe is a very good match to the parameters specified in this inquiry as it is stated above. That word has not yet been suggested (or approached) by any answer or comment (active or deleted) at the time of this edit. This request is changed to a single word request. (However, if you feel a single word fits well but not without some additional supporting words, please offer your thoughts in an answer.) While my curiosity for the term has already been met, the request and bounty remain as an open challenge. I will post my own answer once the bounty has expired and been awarded. Unable to start XAMPP using sudo in Desktop Entry iCloud storage always full GDAL HDF4 Driver on Linux System I'm running code that works with MODIS HDF4 datasets on a Debian Linux distribution.  I am also running this code on a Windows machine.  On both the Windows and the Linux machines, I have the most recent GDAL version (gdal 1.11.1, np19py27_1), however on the Linux machine, GDAL doesn't seem to recognize HDF4 Drivers... it won't load or process any of the HDF4 datasets that are easily processed on the Windows machine (unfortunately this machine doesn't have enough space to run the full code).  Has anyone else encountered this problem?  Do I need to install an additional module on the Linux system? Pokemon Go login issues after update I downloaded the July 30th update for Pokemon Go and when I open the app I'm prompted to login, But once I login, it redirects me to this page on Cookies::     If I click "Done" in the top left, it prompts me to login again! I've tried reinstalling the game, restarting my phone, etc. I can't much enable Cookies on the inline Safari browser. It doesn't make sense, and cookies are on in Safari (because I don't use Safari and they're enabled by default).  How can I fix this? I have a batch file which I use to create an .apk file on my Windows machine. Now I need to be able to create the .apk file in Ubuntu but I don't know how translate my .bat file to a script to be able to run it on Ubuntu.  Below is the batch file which works fine on Windows.  Will you please give me some hints on how I can run it on Ubuntu?  @echo off set PAUSE_ERRORS=0 :user_configuration  :: Path to Flex SDK  set FLEX_SDK=C:\sdk\flex_sdk_4.5.1.21328  :: Path to Android SDK set ANDROID_SDK=C:\sdk\android  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :validation if not exist "%FLEX_SDK%\bin" goto flexsdk if not exist "%ANDROID_SDK%\platform-tools" goto androidsdk goto succeed  :flexsdk echo. echo ERROR: incorrect path to Flex SDK echo. if %PAUSE_ERRORS%==1 pause exit  :androidsdk echo. echo ERROR: incorrect path to Android SDK in 'bat\SetupSDK.bat' echo.  if %PAUSE_ERRORS%==1 pause exit  :succeed set PATH=%PATH%;%FLEX_SDK%\bin set PATH=%PATH%;%ANDROID_SDK%\platform-tools  :: Android packaging set AND_CERT_NAME="PeymanApp" set AND_CERT_PASS=fd set AND_CERT_FILE=cert\SampleApp.p12 set AND_ICONS=icons/android  set AND_SIGNING_OPTIONS=-storetype pkcs12 -keystore "%AND_CERT_FILE%" -storepass     %AND_CERT_PASS%  :: Application descriptor set APP_XML=application.xml  :: Files to package set APP_DIR=bin set FILE_OR_DIR=-C %APP_DIR% .  :: Your application ID (must match &lt;id&gt; of Application descriptor) set APP_ID=air.com.doitflash.SampleApp  :: Output packages set DIST_PATH=dist set DIST_NAME=PeymanApp  :validation %SystemRoot%\System32\find /C "&lt;id&gt;%APP_ID%&lt;/id&gt;" "%APP_XML%" &gt; NUL if errorlevel 1 goto badid goto end_validation  :badid echo. echo ERROR: Application ID (APP_ID) does NOT match Application descriptor '%APP_XML%'     (id)  echo.  :end_validation  set TARGET= set PLATFORM=android  ::call bat\Packager.bat if "%PLATFORM%"=="android" goto android-config  :android-config set CERT_FILE=%AND_CERT_FILE% set SIGNING_OPTIONS=%AND_SIGNING_OPTIONS% set ICONS=%AND_ICONS% set DIST_EXT=apk set TYPE=apk goto start  :start if not exist "%CERT_FILE%" goto certificate :: Output file set FILE_OR_DIR=%FILE_OR_DIR% -C "%ICONS%" . if not exist "%DIST_PATH%" md "%DIST_PATH%" set OUTPUT=%DIST_PATH%\%DIST_NAME%%TARGET%.%DIST_EXT% :: Package  echo true echo. call adt -package -target %TYPE%%TARGET% %OPTIONS% %SIGNING_OPTIONS% "%OUTPUT%"     "%APP_XML%" %FILE_OR_DIR% -extdir lib/ echo. if errorlevel 1 goto failed goto end  :certificate echo Certificate not found: %CERT_FILE% echo.  if %PAUSE_ERRORS%==1 pause exit  :failed echo APK setup creation FAILED.  if %PAUSE_ERRORS%==1 pause exit  :end How can continue the equation of the rectangle red in next line?   \documentclass[11pt,a4paper,twocolumn]{article}  ...  \begin{equation*}     \left| \dot{r} \right| =     \sqrt{{- \, {e}^{-t} \left( \cos t + \sin t \right)}^{2} \, + \, {{e}^{-t} \left( \cos t + \sin t \right)}^{2} \, + \, {(- \, {e}^{-t})}^{2}} \end{equation*} In Oblivion, I had two favorite money-makers:   Dump all my stuff except some basic armor and potions, and raid an oblivion tower. The armor and weapons was worth lots, and selling it in Anvil made a lot of cash fast. Harvest an entire farm's worth of crops, create 'restore fatigue' potions and sell them.   Skyrim seems to spoil this:   Merchants have a max-amount per day/QOD/... instead of per item You can't carry as much (or maybe I have yet to discover the strength/feather spells) Alchemy is a lot harder (most common items go into food recipes now, which don't sell for those prices) On top of that, once a dungeon is cleared, it doesn't respawn - so no 'farming' familiar dungeons?   What are some easy/repeatable money making methods in Skyrim? Which are most efficient effortwise/real-life-time wise? The definite article implying the generic idea of something I know that the definite article "the" can be used to mean a generic idea of something as in "the lion is the king of the animals." In this sentence, the in "the lion" is used to imply the general image of lions, so not specifying a particular lion.   However, I am confused in identifying where a "the" in a sentence is used for a general idea or a specific reference. For example, in the sentence "the price of gas is soaring", I heard that "the" in the sentence is an example of the generic use of the. Although I can easily understand the generic use of "the" in the lion example, but I do not understand how there is an general idea about gas price. Soaring gas price is something real,and gas prices in different regions( in a hypothetical situation) can be different. So, if I say "the price of gas is soaring", do I mean the price of gas that is determined by the world's oil system or can I say "the gas price" even when I am indicating the state of gas price in "a particular region"?    I know it is even difficult to understand my question, but I hope someone could find what I am confused about and give an answer. Is there a way to adjust the torch/flash brightness level on an iOS device?