When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates $(r,\theta,\phi)$, since the problem has rotational symmetry. The solution has the form $$\psi_{lmn}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)$$ where $R(r)$ describes the radial part and $Y_{lm}(\theta,\phi)$ are the spherical harmonics.   When deriving this solution, one normally aligns the problem to the $z$ coordinate and gets a representation for $Y_{lm}(\theta,\phi)$ with the angle $\theta$ measured from the $z$-axis.  One "surprising" observation is, that the spherical harmonics and therefore the solutions of the wavefunction and its absolute squared are not rotational symmetric. I. e. it has these "lumps" of high probability density when forming the absolute squared wave function.  My question is, how does this asymmetry and distinguishing the $z$-axis relate to the "image" of hydrogen? If I actually had a single hydrogen atom and would want o measure the probability density of the electron: To which axis would the wave function be aligned?
In an hydrogen-like atoms the orbitals are solutions to the Schrodinger equation suitable for the problem.  They describe the regions where an electron can be found.  So, why don't they have spherical symmetric?  As in this picture:    Why, in a certain distance from the nucleus, are there points where an electron can be found, and points where it can't?  EDIT: Is there any intuation about the symmetry breaking?
Earth Engine convert list with coordinates and values into a feature collection for export I need to create a feature collection from a list containing coordinates and band values.  These are extracted from a landsat image as shown in this linked script.    I'm unable to find a solution in creating a feature collection.  Here is the tail end code.  var valuesList = joinedImage.reduceRegion({   reducer: ee.Reducer.toList(4),   geometry: myGeometry }).values().get(0);  var feature = ee.Feature(null, valuesList);   // Wrap the Feature in a FeatureCollection for export. var myFeatures = ee.FeatureCollection([feature]);   // Export the image, specifying scale and region. Export.table.toDrive(myFeatures, "data", "myData", "B4-B5",  "CSV");   This doesn't work, because it is clearly the wrong way to create the feature collection.  I've tried to create it similar to Rodrigo E. Principe's solution of mapping a Feature Collection but I can't seem to create the initial feature collection to start with. Is there a way to change the color of text in a question? I just asked this question: .  The first answer I got totally missed the point of the question (that answer is now deleted).    I think my question would have been clearer if I could have changed the color of a (non-code) word to red, so I could show an example of what I wanted.  Is there a way to do this? How to partition my hard drive for Ubuntu use? Till now I was windows user. From now on I want to use only Linux.   I have 500Gb HDD. How do I partition it properly? I read that there is no right or wrong way, but still, I am confused. I did something and I have primary partition mounted on / (160Gb) which I believe is a OS and 350Gb extended partition of which I have 4Gb of swap and 346Gb mounted on /home.  I got used to C:\ and D:\ partitioning, but I don't see file system in that way. I am lost. Where is what? How can I make C:\ partition for OS and D:\ for apps, movies, music, photos. Or what I want is Windows way and I have to get used to Linux way? How do I mount a qcow2 disk image? Assuming $f:\mathbb R\to\mathbb R $ be an uniform continuous function, how to prove $$\exists a,b\in \mathbb R^+ \quad \text{such that}\quad |f(x)|\le a|x|+b.$$ Cross tabulation of two categorical variables: recommended techniques Will my device work with Ubuntu? I have a device on which I would like to install Ubuntu. How do I find out if Ubuntu will work on it? In other words, how to prove:  For all real constants $a$ and $b$ such that $a &gt; 1$, $$\lim_{n\to\infty}\frac{n^b}{a^n} = 0$$  I know the definition of limit but I feel that it's not enough to prove this theorem. How to randomize (shuffle) a JavaScript array? How to use mirror ball to light a scene? I'm trying to figure out how to use "not only... but also" properly. Basically, my goal is to combine two clauses by using "not only".  For negations, I've figured out two styles that both sound correct:   He knew that if he fractures his finger, not only would he not be able to compete in the water polo tournament, he would not be able to take the SAT on Monday, either. He knew that if he fractures his finger, neither would he be able to compete in the water polo tournament, nor would he be able to take the SAT on Monday.   However, I don't know how to form a sentence that does not include a negation while using "not only". Here are a few possibilities that I've considered:   Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, but he was also given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, but he was given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, he was given a full scholarship. Thanks to his remarkable performance on the SAT, not only was he accepted into Harvard, he was also given a full scholarship.   In those four possibilities, I'm varying the usage of "but" and "also" in the last clause.  What is the proper usage of "not only"? Bug Identification: Tiny Bugs I have seen these on my hands recently and I am wondering what they are. Urban Setting. South Central Pennsylvania. $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$ Auto-implemented getters and setters vs. public fields I see a lot of example code for C# classes that does this:  public class Point {     public int x { get; set; }     public int y { get; set; } }   Or, in older code, the same with an explicit private backing value and without the new auto-implemented properties:  public class Point {     private int _x;     private int _y;      public int x {         get { return _x; }         set { _x = value; }     }      public int y {         get { return _y; }         set { _y = value; }     } }   My question is why. Is there any functional difference between doing the above and just making these members public fields, like below?  public class Point {     public int x;     public int y; }   To be clear, I understand the value of getters and setters when you need to do some translation of the underlying data. But in cases where you're just passing the values through, it seems needlessly verbose. Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa? Thanks to the internet, graphic design and programming are becoming increasingly overlapping fields.   As a programmer, I constantly find myself needing basic knowledge of Graphic Design, but have trouble locating thorough yet concise graphic design learning resources, and instead usually find bits and pieces of knowledge applicable to the task at hand.  Can you guys offer some tips or recommend some good books, tutorials, blogs, and websites which provide a beginner level overview of web design that have been, in your experience, effective and high quality learning materials. I'm not looking for tips on how to use image editing tools, but assuming a basic grasp of those, which resources and advice will best help a new designer create professional looking designs? Locate the ".Rprofile" file generating default options How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any advanced/professional techniques. It can be proved using only basic techniques. My first attempt was to assume that the series does converge to a rational number $ \frac{p}{q}$ and then it follows by definition that for every positive epsilon there exists an integer k such that $ {\sum_{n=1}^{k} \frac{1}{(n!)^2} -\frac{p}{q}}&lt;=\epsilon$ I tried to get a contradiction but I failed. In the PHP manual, to show the syntax for functions with optional parameters, they use brackets around each set of dependent optional parameter. For example, for the  function, the manual reads:  string date ( string $format [, int $timestamp = time() ] )   Where $timestamp is an optional parameter, and when left blank it defaults to the  function's return value.  How do you go about creating optional parameters like this when defining a custom function in PHP? Skipping acquire of configured file 'main/binary-i386/Packages' as repository 'xxx' doesn't support architecture 'i386' Getting below error message while issuing :  sudo apt-get update  Get:1 http://us.archive.ubuntu.com/ubuntu xenial InRelease [95.8 kB] Ign:2 http://repo.mongodb.org/apt/debian wheezy/mongodb-org/3.2 InRelease                                                                                                             Ign:3 http://dl.google.com/linux/chrome/deb stable InRelease                                                                                                                          Hit:4 http://ppa.launchpad.net/canonical-x/vulkan/ubuntu xenial InRelease                                                                                                 Hit:5 http://repo.mongodb.org/apt/debian wheezy/mongodb-org/3.2 Release                                                                             Hit:6 http://us.archive.ubuntu.com/ubuntu xenial-security InRelease                                                   Ign:7 http://dl.google.com/linux/talkplugin/deb stable InRelease                                                      Hit:8 http://us.archive.ubuntu.com/ubuntu xenial-updates InRelease                                                    Ign:9 http://linux.dropbox.com/ubuntu wily InRelease                                            Hit:10 http://ppa.launchpad.net/numix/ppa/ubuntu xenial InRelease                               Get:12 http://dl.google.com/linux/chrome/deb stable Release [782 B]       Hit:13 http://dl.google.com/linux/talkplugin/deb stable Release                   Ign:14 https://apt.dockerproject.org/repo ubuntu-wily InRelease           Hit:15 https://apt.dockerproject.org/repo ubuntu-wily Release             Get:16 http://dl.google.com/linux/chrome/deb stable Release.gpg [181 B]   Hit:17 http://linux.dropbox.com/ubuntu wily Release                                   Get:20 http://dl.google.com/linux/chrome/deb stable/main amd64 Packages [1,191 B] Fetched 98.0 kB in 0s (118 kB/s)                                  Reading package lists... Done N: Skipping acquire of configured file 'main/binary-i386/Packages' as repository 'http://dl.google.com/linux/chrome/deb stable InRelease' doesn't support architecture 'i386'   This cuts across various releases of ubuntu