@BernardMeurer You know better than that. Those things are only "low power" by comparison to very modern machines. I'd have killed for a machine with those specs in grad school.
@dmckee on a serious note though: 1. I only did that joke because I know you're one of the very few here who know what a PDP-11 is 2. Compiling things is hard even for powerful machines. Specially larger packages, try compiling a GNU toolchain and feel the bern
so the way it works is your system compiler compiles a version of GCC capable of compiling the full GCC, that version of GCC compiles full GCC, then full GCC compiles itself
@HDE226868 you mean this one? that's roughly what was in the original version, before I was successfully steered away from a bar chart (which in retrospect is indeed rubbish). I'm not sure the horizontal axis is that well used in the new version, though.
On a similar track, though, I recently heard this from a PI here. Back in the late 80s / early 90s TeX was already a thing, but postscript displays were not. So he had a secretary to type papers up, except that instead of a typewriter she would TeX it up on a computer, and to see the output she needed to use a postscript printer.
user218912
@BernardMeurer from my understanding they copied answers from an older lab onto a new lab, but the questions weren't exactly identical and they didn't take that into consideration when copying.
@EmilioPisanty I remember using TeX before there were postscript printers. We had to print out on a dot matrix printer. But it was desperately slow, and not great quality.
@GPhys all I'm saying is - be prepared to be told by others that it's not clear-cut. When you're doing $f:A\to B$ it's not clear at all whether it's punctuation or a relation.
@EmilioPisanty (forgive me for responding) I definitely concede to your general point that there's variation in preference and no objectively correct usage, but I simply mean to say there's often standard choices as suggested by how they were designed or in common use today
@GPhys as I said, I disagree. There's plenty of "standard choices" which simply come from the tex old guard's aesthetic and nothing more. In this specific case, it doesn't even match current usage - check your textbooks and you'll see that \colon has a much tighter left-hand spacing than most common texts. But as is usually the case once a tex purist gets going there's no arguing with them, so feel free to berate your peers with your canonical opinions on matters of taste.
Let's try it: The general formula for the period of a 1D motion is $T(E) = \sqrt{2m}\int_{x_1}^{x_2} \frac{\mathrm{d}x}{\sqrt{E - V(x)}}$, where $x_1,x_2$ are the turning points and $E$ is the total energy of the system.
For a symmetric potential, $x_1 = -x_2$ and $E = V(x_2) = V(-x_1)$.
Wait, how do you expect "multiples" of a base frequency at all, @DanielSank? The period of oscillation is either constant or a continuous function of the energy.
@DanielSank yeah, it's not quite clear what you mean there
Say you have a softened Coulomb potential in 1D QM and you're looking at the oscillations of the expected position given an arbitrary bound initial state, then the frequency spectrum of the oscillations will include all frequency differences of the well.
On the other hand, if you have a reflection-symmetric system responding to a sinusoidal driving with period $T=2\pi/\omega$ and you neglect transients, then yes. If you delay the forcing by $T/2$ and you reflect the system and the driving about the plane of symmetry, then you're left with exactly the same system and exactly the same driving.
Since you're neglecting transients, this means that the response needs to be the same. That rules out even harmonics, which would be flipped by the procedure.
When two waves fully constructively interfere, is it not the case that the amplitude of the wave doubles (assuming the waves are identical in all regards)?
@ACuriousMind @EmilioPisanty ok here's what I really want to know...
I heard some vague statement that for a given oscillating system with coordinate $A(t)$ and frequency spectrum $\tilde{A}(\omega)$, there is some kind of correspondence $\tilde{A}(\omega) \rightarrow \langle \psi | \hat{A} | \phi \rangle$ if $E_\psi - E_\phi = \hbar \omega$.
I want to understand that so I can try to understand if there's some classical intuition behind selection rules.
But your usage of "frequency spectrum" implies that you are not talking about about the frequency/period of the motion itself, but about the frequencies that occur in the Fourier transform of the motion.
@ACuriousMind Well, yes. That's the first frequency component. In general, the Fourier transform of $A(t)$ has a component at $2\pi n / T$ for all integers $n$.
Anyway, if you're talking about the Fourier spectrum, then the initial claim you asked about follows from the solution of the motion being purely odd in the time domain (unless you have symmetry breaking, the motion in an even potential will be odd if normalised to $x(t = 0) = 0$) and odd functions have odd Fourier transforms.
So if I'm reading it correctly, the precise claim in LL is this. Say you have a 1D classical system, and you quantize its classical orbits via $\frac{1}{2\pi\hbar}\oint p \mathrm dx= n+\frac12$. Then for large $n$ the energy spacing $\Delta E$ betwee two consecutive levels is $\Delta E = 2\pi \hbar /T$, where $T=T_n\approx T_{n+1}$ is the period of the motion on the levels in question, which is assumed not to change much.
They also claim that it's a good approximation to set $f_{n+s,n}\to f_s$, where $n$ is taken at some representative point in the band. They claim this is a good approximation because in this regime $f_{m,n}$ varies much more strongly with $(n+m)/2$ than it does with $m-n$.
That seems reasonable to me under the given assumptions.
Under those conditions, the expectation value reduces to $f=\sum_s f_s e^{i\omega s t}$
that looks like the Fourier transform of some periodic function
> Since $f$ must, in the limit, coincide with the classical quantity $f(t)$, we arrive at the result that the matrix elements $f_{mn}$ in the limit become the components $f_{m-n}$ in the expansion of the classical function $f(t)$ as a Fourier series.
That seems reasonable to me in as much it is an analysis of what happens to the quantum description in the classical WKB limit of $n\to\infty$, $\Delta n \ll n$.
@heather What Dan mentioned is important, the notation he did would mean for you to run the command blah blah (which doesn't exist afaik, although you can try, lol)
@heather Yes, notation, the "websites" (correct name, servers), that hold the packages are to be called "sources", "repositories" is also used, short "repo"
@heather Thus, when adding a new server to that list you could say you added a new source. In Debian-based distributions only you may also (and it is common) call them a PPA
@DanielSank You can weaken that a bit, though, by making it a weighed average of the $⟨n+s|\hat f|n⟩$, with weighing through the $a$'s.... I think?
@DanielSank summed out to $1$
but now that I look at it, it looks increasingly dodgy
they go from $f =\sum_{n,s} a_{n+s}^*a_n f_{n+s,n} e^{i\omega s t}$ to $$f=\sum_{n,s} a_n^* a_n f_s e^{i\omega s t} = \sum_n |a_n|^2 \sum_s f_s e^{i\omega s t} = \sum_s f_s e^{i\omega s t}$$
> The matrix elements $f_{mn}$ calculated by means of the quasi-classical wave functions decrease rapidly in magnitude as the difference $m-n$ increases, though at the same time they vary only slowly with $n$ itself ($m-n$ being fixed). Hence we can write approximately...
so.... essentially they're saying it's OK to swap out $a_{n+s}^*$ for $a_n^*$?
@BernardMeurer, no, because it needs sudo, and I thought you were debating over whether or not entering the command was a good idea. I can enter it though if you want
@heather Interesting thing. To cancel a command in the terminal, you can do ctrl-c. So in order to copy something from the terminal (which you must select with your mouse afaik), you must do ctrl-shift-c
@heather I gotta attend to some matter now, shouldn't take long. @DanielSank After she runs that command, teach her: 1. About not running commands she doesn't understand. It's like talking to strangers 2. UNIX permissions, that `stat` should yield the perms of the sources file
