I've searched around here and it seems I'm not alone with my frustration towards Griffith's explanation of degenerate perturbation theory. Right from the start, Griffiths claims that some perturbation $H'$ will split a degeneracy (assume 2-fold degeneracy) and if we 'turn off' the perturbation then the 'upper' and 'lower' state will be some mutually orthogonal linear combinations of the degenerate states $\psi_A$ and $\psi_B$. Where did this claim come from? Is this because the linear sum $c_1\psi_A+c_2\psi_B$ is also an eigenstate of the unperturbed Hamiltonian $H^0$?   My intuition so far is that we resort to linear sums because we can now have distinct wave functions to use in the first order correction $E_1^1=\langle \psi_A^0 |H'|\psi_A^0 \rangle$ or $E_1^1=\langle \psi_B^0 |H'|\psi_B^0 \rangle$; I guess it was unclear which wave function we must choose? Is this why Griffiths goes on to solve $H\psi^0=E^0\psi^0$ where $H=H'+H^0$ and $\psi^0=c_1\psi_A+c_2\psi_B$?   I found some more formal mathematical reasoning behind this. I can't understand the explanations as I don't have a very advanced understanding of linear algebra, but from what I get we are trying to change basis? Can I get some clarification on the mechanism and reasoning behind this?
I have for quite some time now tried to find a satisfactory answer to this, but I haven't yet. In perturbation theory, with small parameter $\lambda$, we expand the eigenstate as  $$| E \rangle=| E^{(0)} \rangle + \lambda | b \rangle + ...$$  Where $| E^{(0)} \rangle$ is an eigenstate of the unperturbed Hamiltonian. The problem is that when there is degeneracy, there is a choice of these eigenstates. I know the answer is that they are chosen to be eigenvectors of the perturbing Hamiltonian, but my question is why.
How to space columns evenly in a table with multicolumns? Find index of nested item in python Why is a (simple) mask not recommended for asymptomatic COVID-19 patients? Information about (simple) mask wearing (concerning covid19) is a complete mystery to me. On several official places it is not recommended for a person without symptoms to wear a mask. For example:    (time 0:20)  or here:    However no explanation is given - which makes me wonder (and even angry). I think everybody should wear a mask because:   Many persons feel ashamed wearing it, even those with symptoms. If everybody wears it, they do not feel ashamed and wear it. As far as I know, one is (wrt covid19) contagious before developing symptoms. So if everybody wears the mask also these people do.   The two above points act as a collective protection.   Simple mask certainly will lower "getting / spreading illness" probability. Of course the "holes" (in a simple mask) are too big to prevent completely virus to enter (or get out when coughing), but it lowers probability! Even if it lowers the transmission probability by e.g. 5%, it might be a lot. As a hypothetical example: it might reduce R0 from 1.02 to 0.97 and so stop the virus spread so fast.   The videos are full of self-contradiction. In the same video    time 0:50 the lady says I should wear a mask if I am in contact with an infected person... why if it is inefficient?  Question: What are reasons for not recommending wearing a mask? bash: syntax error near unexpected token `}' This is a phrase I've heard many people use, and it sounds wrong to me; e.g.:     The thing about that is, is that she might take it the wrong way.   It seems to treat "The thing [...] is"—the entire phrase—as the subject of the sentence. Of course, even if that were allowed, the comma placement would still be questionable.  I'm curious to know:   Is there a name for this wording? Is it specific to a certain dialect? Is its etymology known? Additive group contains exactly three elements of the set $\{p,p+q,pq,p^q,q^p\}$ Let $p$ and $q$ be distinct primes. There is a proper subgroup $J$ of the additive group of integers which contains exactly three elements of the set $\{p,p+q,pq,p^q,q^p\}$. Which three elements are in $J$?  I know the solution is $\{p,pq,p^q\}$. However I do not understand why. I thought $J$ can not even be a finite group. Maybe there is serious misunderstanding of the problem. Can anyone give me a hand? Thanks. How to create a desktop shortcut in Unity? Is there an easy way to create a desktop shortcut for an application in Unity? The ability to right click or drag and drop icons to the desktop in Unity does not seem to be there. In Ubuntu 20.04, on the desktop, there is a number of actions I wish I could do, such as, by order of importance :  use keyboard shortcuts such as select all, copy / paste. use drag &amp; drop with icons deactivate the &quot;mouse over&quot; highlight, and restrict the clicking area to only the icon, not the surrounding square  To put it simply, I'd like to have the exact same features as offered by the desktop of Ubuntu 18.04 (probably related to a specific version of GNOME, and not Ubuntu itself). Am I missing something obvious here? I actually see no reason why those actions are removed or deactivated on the desktop, since they all work in any Nautilus window. I'm writing an article on  and some people pointed out that this is completely useless.  My idea is to amend my article with some theories that seemed useless when they are created but found use after some time.  I came with some ideas like the Turing machine but I think I'm not grasping the right examples.  Can someone point me some theories that seemed like the Lychrel numbers and then become 'useful'? I have an iPhone 5 running iOS 9.1 and a Macbook Pro running Yosemite.  Both devices are set up to sync photos to and from iCloud.  When I take a photo on the iPhone (for further processing on the Macbook), it takes an inordinate amount of time until the photo shows up in the Macbook's photos library - sometimes up to 10-20 minutes. Sometimes it seems to work faster if I quit the Photos app on both devices, and start it again.  I often end up sending myself photos through E-Mail because it's quicker. That seems ridiculous!  Is there a way to force a sync on the iPhone, or to speed up the process some other way? How to prove a total order has a unique minimal element Let $R$ be a total order on set $S$. Prove that if $S$ has a minimal element, than the minimum element is unique.  I have difficulties with proofs. I know any graph of a total order is a straight line, which clearly has a minimal element. How doI tell when something is mathematically proven?  EDIT: this is how I would prove it  Let $x, y \in S$ such that x and y are minimal. Since S is a total order $x$ and $y$ are comparable and either    case 1: one comes after the other and therefore one isn't the minimal case 2: both are the same and $x=y$  I know this isn't a mathy argument and could use help making it more presentable. Find highest level of a hierarchical field: with vs without CTEs is a dove a "she"? how to use "he" and "she" for animals? In the song &quot;Blowing in the Wind&quot; by Bob Dylan, there's a verse that reads:  Yes, and how many seas must a white dove sail Before she sleeps in the sand?  My question is, why he used &quot;she&quot; for a dove? What about other animals and other non-humans? Is there a general rule? Scheduling a Round Robin tournament - 4-way games I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table.  I'd like for each player to play with each other player exactly once over the course of 5 rounds.  I just took a look at this post:  , and I was wondering if anyone knows if the case n=m=4 has been solved, and if so what that solution is. Dear all I think I have a problem with the collation of makeidx.  following some suggestions found here I have got in the text something like this  \index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|qt}  \index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00868 @868f-869b|nn{\thefootnote}}   the first is an \index entry in the text, the second in a footnote. My problem is that even if I put the sort also on the second level of indexing it keeps keeping them separated.  I use this package information  \usepackage[splitindex]{imakeidx} \makeindex[name=n, title=Nomi di persone e luoghi]  \makeindex[name=p, title=Passi citati nel testo] \makeindex[name=t, title=Altri testi citati] \makeindex[name=pap, title=Papiri e manoscritti citati nel testo]  \newcommand{\nn}[2]{#2 \textit{n}#1} \newcommand{\qt}[1]{\textbf{#1}}   in the .idx file I get this  \indexentry{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|hyperindexformat{\qt}}{8}  \indexentry{Plutarco!demalignitate@\textit  {De Malignitate Herodoti}!00868 @868 b-c|hyperindexformat{\nn{33}}}{10}   can somebody advise me on what I am doing wrong here, please? I have tried to use \string after the first sort with no effect.    sure, sorry: this is the file I'm using  \documentclass[12pt,a4paper]{book} \usepackage[utf8x]{inputenx}  \usepackage{palatino}        \usepackage[LGR,T1]{fontenc} \usepackage[english,german,polutonikogreek,italian]{babel}  \usepackage[splitindex]{imakeidx} \makeindex[name=p, title=Passi citati nel testo,columns=1] \newcommand{\nn}[2]{#2 \textit{n}#1} \newcommand{\qt}[1]{\textbf{#1}} \usepackage[hidelinks]{hyperref} \begin{document}  (855A\index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|qt})% \footnote{DHM 868 B-C\index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00868 @868 b-c|nn{\thefootnote}}} \printindex[p] \end{document} Understanding the Durbin Watson test I'm searching for a word to indicate something that was current at some time in the past, but not necessarily now.    For example:       The doctor's report of June 2008 set forth the patient's ?{then-current} complaints. Do you need to erase the memory of *watching* a memory in a Pensieve? fp - How to keep running sum in variable? I have this code in my LaTeX:   \FPset{totalHours}{0}  \newcommand{\entry}[5]{     \FPeval{totalHours}{(totalHours)+#5}     #1 &amp; #2 &amp; #3 &amp; #4 &amp; #5 \\ }   Where \entry is meant to be an entry in a time log. The 5th argument is a number representing the number of hours in that entry. I want to keep a running total of the hours in a variable I can print later on, totalHours.   This seems to do the \FPeval but then totalHours gets reset to 0 by the next time I write \entry. What is the correct way to keep a running total? Thanks. JavaScript: Bitwise shift of long long number I need to bitwise shift a value 64 times in JavaScript. But JavaScript starts rounding after 32.  For example:  for(var j = 0; j &lt; 64; j++) {     mask = mask &lt;&lt; 1;     console.log(mask); }   This prints value from 0 to 1073741824 but then rounds of and starts printing 0.