I'm reading from the "Modern Physics" textbook by Randy Harris.  I'm Chapter 8 on Spin and Atomic Physics, and the book has just introduced how to solve for the wave function of 2 particles in a box. The two particles are identical, and can have different quantum numbers. They do not interact.  From there they go on to explain why we use symmetric and antisymmetric wave functions, because we can't be able to tell which particle is which from the probability density equation, which we ARE able to without using the symmetrical requirement if the two particles have different quantum numbers.  My question is, why can't we distinguish between two particles with different quantum numbers? Doesn't that mean they have different energies, and as such have different wave forms and so appear in different places? The problem used in the book is that the two particles are in a box. But because of their different quantum numbers, one of the two particles can't be found in the middle of the box, while the other can. This should never be the case apparently, so they introduce the symmetrical requirement.  As stated above, I'm confused as to why we're not supposed to be able to tell two particles with different quantum numbers apart.  Thanks for your time.
I've been going through Shankar's Principles of Quantum Mechanics. In the section of the system of identical particles, he uses an example of billiards to illustrate the difference between identical particles in classical vs. quantum mechanics. He argues that in classical mechanics, we can track the history of a particle (a billiard ball) to distinguish it from another particle with no intrinsic differences. In quantum mechanics, however, he argues since continual observation is not possible, we can't use the same method to distinguish identical particles. A potential counter-example I thought was: suppose we have two non-interacting particles in the same square well. And at the end of some measurement, we find that one particle 1 is in a stationary state $\psi_1$, and particle 2 in $\psi_2$. We measure the system again after some time $t$, then we know whichever particle that's in $\psi_1$ must be particle 1 from the previous measurement. And the same goes for particle 2. Thus we can distinguish the two &quot;identical&quot; particles. What conceptual mistakes am I making here?
Limit with multiple variables I need to evaluate the limit $$\lim_{n\rightarrow \infty}(n-\sqrt{n+\alpha}\sqrt{n+\beta})$$  My work: $$(n-\sqrt{n+\alpha}\sqrt{n+\beta})=n-\sqrt{n^2+\alpha n+\beta n+\alpha\beta}$$ I understand that both $\alpha$ and $\beta$ are just constants. But, now how should I factor this? I understand the calculus behind limits, but the algebra troubles me. How to use the symbol \angle at the section title Where did the convention of using single dashes for letters and doubles dashes for words come from and why is continued to be used?  For example if I type in ls --help, you see:    -a, --all                  do not ignore entries starting with .   -A, --almost-all           do not list implied . and ..       --author               with -l, print the author of each file   -b, --escape               print octal escapes for nongraphic characters       --block-size=SIZE      use SIZE-byte blocks   -B, --ignore-backups       do not list implied entries ending with ~ ...   I tried googling - and -- convention even with quotes with little success. How can I keep low resolution textures pixelated? Expected Value Of Number of Removals From Urn Here's the question: "An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X."  I'm having trouble finding a closed form expression for the expected value. Here's the work I've done: Can we say that $\frac{0}{0}$ is every number? Conditional \endinput I want to prevent processing rest part of a package in case if it is loaded not from lualatex. For some reason I cannot load ifluatex package, so I borrowed definition of \ifluatex from that package and invented the following chunk of code:  \expandafter\ifx\csname luatexversion\endcsname\relax % = not \inluatex   %\typeout{^^J********************** not LuaTeX^^J}   \PackageError{babel}{     ^^J\space\space * Option russianu is intended for LuaTeX only.              ^^J\space\space * Option russian  is used instead.                     }   \relax   \input{russianb.ldf}     \let\@next@action\endinput \else   %\typein{^^J********************** LuaTeX^^J}   \let\@next@action\relax \fi \@next@action   It seemed to work but Is there a better solution. In particular, it is desirable to avoid introduction of \@next@action because it is used the only time.  Just to explain: my goal is to hack babel with option russian to make it working painlessly with LuaLaTeX without any additional hacks (as described, eg, in  and ) How is the equation of motion on an ellipse derived? Why are there no animals in "The Matrix"? If one is asking for a person, what should s/he say&mdash;1 or 2?   Could I speak to Kristina? Could I speak with Kristina? Need Help Fixing Grub after Installing Ubuntu 13.04 x64 Alongside Windows 8 on a Lenovo IdeaPad Y400 with UEFI I have a Lenovo IdeaPad Y400 that came pre-installed with Windows 8. I would like to run Ubuntu alongside it in a dual-boot configuration. I've spent the past week fiddling around, re-installing both Windows and Linux in various configurations, and I've come awfully close to success, but seem to have hit a snag with Grub.  For my most recent attempt, I took the following steps:   Disabled Secure Boot Re-imaged Windows 8 from a hidden recovery partition, effectively restoring the machine to factory state Activated an option called Legacy Support in the BIOS (or is it UEFI now?) Booted Ubuntu 13.04 x64 from a USB stick, chose "install alongside Windows" and accepted all default options After a restart, found that I could boot Ubuntu, but could not boot Windows Once into Ubuntu, I ran Boot Repair with recommended settings and accepted all defaults. When Boot Repair finished, I was presented with the following message:      The boot files of [The OS now in use - Ubuntu 13.04] are far from the   start of the disk. Your BIOS may not detect them. You may want to   retry after creating a /boot partition (EXT4, >200MB, start of the   disk). This can be performed via tools such as gParted. Then select   this partition via the [Separate /boot partition:] option of [Boot   Repair]. ()    I followed the instructions at the url in the message, creating a 1GB EXT 4 partition near the start of the drive  when I tried to run Boot Repair again, the "Separate /boot partition" option on the "Grub Location" tab was disabled.  The new partition is /dev/sda11 and has the label GRUB. On restart, I get the following error message:      error: file '/boot/grub/i386-pc/normal.mod' not found.    grub rescue>    However, if I manually choose the boot partition by hitting F12, I can boot into Windows no problems. Selecting Ubuntu from the boot device menu gives me a Grub menu with the following options:      Ubuntu      Advanced options for Ubuntu       Windows UEFI recovery bootmgfw.efi      Windows Boot UEFI recovery       Windows UEFI recovery LrsBootmgr.efi      Windows Boot UEFI recovery bkpbootx64.efi       Windows Recovery Environment (loader) (on /dev/sda3)       Windows 8 (loader) (on /dev/sda5) System setup    Selecting Ubuntu from this menu allows me to get to my Ubuntu desktop. I haven't tried the other options yet for fear of losing the progress that I've made.   So it looks like I have successfully dual-booted the machine, but that Grub is broken for some reason or another.   My current partition scheme can be seen at:   Thanks in advance - Jonathan Are human chromosomes connected or separate molecules? Do the 46 human chromosomes form a single unbroken DNA helix?  Or is it rather that a human's genome consists of 46 disconnected helices?    If it is the former, does the common numbering scheme for the chromosomes have any correlation to their actual ordering in the one large strand?  If is the latter, is there a convention on how the chromosomes are ordered in genomic datasets?  Also, is there a clear understanding of how sister chromosomes "find" each other in Meiosis I?  Generally, during periods when Mitosis/Meiosis are not occurring,  what's a good physical picture for how the chromosomes are physically arranged (e.g. a bowl of 46 spaghetti noodles, or maybe the sister chromosomes always stay close together, etc)  thanks! LaTeX/BibTex not arranging citations by order of appearance I am having an issue with LaTex/Bibtex presenting my citations in order of appearance. I know there a plenty of questions on this and I have read some of them, but the solutions posted don't seem to do me any good. I am using MacTeX and TeXmaker.  I am writing a large document, like a Thesis. Chapters are split between individual Tex files and included using the \include command.  I have a large .bib file that is maintained by Mendeley (program I use for storing my references and producing a BibTex file).  I am using the package, cite (mainly for doing references like [1-3,6,9])  I have a pretty standard looking document.  \documentclass[11pt,a4paper]{report} \setcounter{secnumdepth}{3} \setcounter{tocdepth}{3} \newcommand{\doubleSignature}[3][Alex Mason]{% \parbox{\textwidth}{ \centering #3 \today\\ \vspace{2cm}  \parbox{7cm}{   \centering   \rule{6cm}{1pt}\\    #1  } \hfill \parbox{7cm}{   \centering   \rule{6cm}{1pt}\\   #2 } } } \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} %\usepackage[sorting=none]{biblatex} %\bibliography{Alex} % Where journals.bib and phd-references.bi \usepackage[pdftex]{graphicx} \usepackage{enumerate} \usepackage{nomencl} \usepackage{wasysym} \usepackage{booktabs} \usepackage{cite} \usepackage[left=1in,right=1in,top=1in,bottom=1in]{geometry} \graphicspath{{./Diagrams/}} \usepackage[labelfont={sf,bf}, margin=1cm]{caption} \usepackage{subcaption} \renewcommand{\figurename}{Fig.} \renewcommand{\tablename}{Tab.} \renewcommand{\bibname}{References} \author{NAME (MEng)} \title{TITLE} \date{\today} \makenomenclature %------------------------------------------------ \begin{document} \maketitle \pagenumbering{roman} %-----------------------------------------------certificate of originality----------------      -------------------------------------------- \input{CertofOrigin} \begin{abstract} ABSTRACT \end{abstract} %-----------------------------------------------contents------------------------------------------------------------ \tableofcontents \listoffigures \addcontentsline{toc}{chapter}{List of Figures} \listoftables \addcontentsline{toc}{chapter}{List of Tables} \chapter*{Acknowledgements} \addcontentsline{toc}{chapter}{Acknowledgements} ACKNOWLEDGEMENTS \printnomenclature[1.5in] \addcontentsline{toc}{chapter}{Nomenclature} \pagebreak \pagenumbering{arabic} \include{2ndYearReport_CHAP1} \include{2ndYearReport_CHAP2} \include{2ndYearReport_CHAP3} \bibliographystyle{unsrt} \bibliography{Alex} %\printbibliography \end{document}   The goofy looking part at the top (\newcommand{doubleSig.....) is just for when i input a special page I made containing a certificate of originality spiel with two boxes for signatures and my university logo.  As you can see I am using the standard bibtex way of doing bibliographies by stating the style then the stating the bib file to use right at the end of the document. I have tried biblatex but can't get it to work. It says it hasn't produced the bib file but doesn't really state why.  As far as I am aware everything was working A-OK until I started on the second chapter and began adding citations in there. Now my citations are all messed up and won't correct. My first citation starts with [9]!!!  [1] appears in a figure caption later down the page, after 10, 11 and 12 have been called!  I have tried deleting all the working files (bbl, aux, log etc) but nothing happens. I get the same result. IIRC I used to have this problem with MS Word and is part of the reason why I switched to LaTex...yet it seems to have followed me!   bibtex does through a few warnings when it runs stating that some references are missing journal names or volume numbers. Would that be a reason?  What have I done wrong with my code? Is there a limit to the number of Power Armor sets you can keep? I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep? Do we still need to avoid using frames and iframes for SEO? I thought frame and iframe were bad for SEO, but today I was doing a search for "numismatica" on Google.it.  The 3rd site is lemonete.com which is all made with frames. A competitor site (nummus.com) that has no frames ranks much lower when searching for the same word: "numismatica".  So can we use frames without taking an SEO hit now?  Any explanation (possibly simple) would be appreciated also about the fact the 2nd site ranks much lower. :) Move the Add Favorite / Ignored Tags boxes to the top It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. Two players, $A$ and $B$, alternately and independently flip a coin and the first player to get a head wins. Assume player $A$ flips first.  If the coin is fair, what is the probability that $A$ wins?   So $A$ only flips on odd tosses. So the probability of winning would be $$ P =\frac{1}{2}+\left(\frac{1}{2} \right)^{2} \frac{1}{2} + \cdots+ \left(\frac{1}{2} \right)^{2n} \frac{1}{2}$$  Is that right? It seems that if $A$ only flips on odd tosses, this shouldn't matter. Either $A$ can win on his first toss, his second toss, ...., or his $n^{th}$ toss. So the third flip of the coin is actually $A$'s second toss. So shouldn't it be $$P = \frac{1}{2} + \left(\frac{1}{2} \right)^{2} + \left(\frac{1}{2} \right)^{3} + \cdots$$ Class object affecting other class object when thread.sleep- Python Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $f(f(x) + y) = f(x^2 - y) + 4yf(x)$. Determine all $f : \mathbb R \rightarrow \mathbb R$ such that $$f\big(f(x) + y\big) = f(x^2 - y) + 4yf(x)$$ for all $x,y\in\mathbb{R}$.   $f(x) = 0$ is obvious , but how can we find all other functions?  Thanks in advance! Find highest level of a hierarchical field: with vs without CTEs