Is it a misnomer when an electrostatics problem specifies a conductor to have $\rho$ distributed uniformly throughout

shouldn't it just be $\sigma$ spread across the surface area

\sigma for surface charge

oh right

and you're more-or-less right i'd say---i'd say it's just "charge" which is uniformly distributed, and for a surface that's a uniform sigma

hmm okay. So the picture should still be charge on the surface. It might just be a way for the question to not "give away" that fact.

right

or shorthand for it

sometimes we get so used to saying it the quick way that we forget that it's not obvious

Even in the classical picture, it's not implied that $\mathbf{E}=0$ in a conductor btw

well, you do have to say something about electrostatic equilibrium

it has to be assumed since it's not impossible for charge to exist along the axes/points of symmetry for the conductor

i.e., a single point charge in the center of a sphere, or a continuous line of charge along the cylindrical axis of symmetry for a cylindrical conductor

true

there's probably some semantics about what the right 'definition' is

the formal definition i'd probably fall back on is "a conductor is what you get if you start with a dielectric material and take $\epsilon\to \infty$."

@SillyGoose

oh i think the implication was that it wasn't a conductor but an insulator btw

@RyderRude the domain of the functor is the group. It isn't the category of groups

Any group is a category with a single object

@Obliv in reality it is very logical in electrostatics. The definition of conductor is a material where charge is free to move, at equilibrium (which is were electrostatics works) no charge should be moving and for that to happen you need $\vec{E} = 0$ inside the conductor or else the charge would be free to move

For the level of the discourse it is extremely clear that it is some kind of misuderstanding. It should be an insulator. But when you learn further and need to model what the electrons inside a metal is doing, then we would treat all the freed electrons in a statistical thermodynamics model, separate from all the ions. The first approximation is then to smear the ions out into a uniform charge density background. Later, when you learn crystalline physics, then you can upgrade to crystalline

ionic background

@Relativisticcucumber amazing

XD sidney coleman just lighting up a cigarette at the beginning of lecture: youtube.com/watch?v=mnlHLd3UtYA

@alam Hi :-)

Hi:)

Can I move this to the problem solving room? We prefer the chat here to be about general principles rather than specific problems.

Ok

→ 5 messages moved to Problem Solving Strategies

is this an appropriate writing down of the Hamiltonian of an ideal scattering experiment?

i remark that I do not explicitly define what the Hamiltonian is between the asymptotic behavior and the bounds of the finite interaction interval $\mathcal{I}$

@alam hi

@VincentThacker so the map between objects is just $F(SO(2))=R^2$, right? I'm using $SO(2)$ to denote the one object of the group

i understand the map between morphisms.. the map between objects is weird becuz ive never seen a map of this form

i am also wondering if this is an appropriate interpretation of the $S$ matirx

in QFT we have these dirac deltas pop up that also encode conservation of (insert quantity) to my understanding

bleb well i think sakurai is saying that $1 - blah$ is the amplitude for no scattering and $blah$ is the amplitude for scattering

@SillyGoose yes

i don't understand how this quantity can be a distribution? at what point did we turn it into a distribution... it should just be an inner product i would have thought and so a complex number

but it is expected to be a distribution becuz it shud b delta in the absence of scattering

i should clarify that the incoming and outgoing state at this point is assumed to be a particle in a length $L$ box for now, but i guess i see your point once we take the $L \to \infty$ limit

okay i guess that makes sense. if the incoming and outgoing waves are plane waves then surely the $S$-matrix is not a discretely indexed matrix (in the momentum basis)

okay so it is really an artifact of the idealized set up

and then when we move into a real picture maybe we integrate twice: once over $k$ and once over $k'$ to get our packets

but the momentum space is discrete for qft in a box

i guess it depends on how u r setting things up

hm well i do have momentum space as being discrete at this point

so i have the kronecker delta indexed by $k$, and the dirac delta just appears from evaluating the integral in (72)

oh

but i think what you said makes sense bc ultimately i will take $L \to \infty$

that is how sakurai sets it up

so the $S$-matrix here is really like an $S$-matrix density or something

@RyderRude Yes

Is there any modern physics book which is not just plug the numbers and get the answer( I have books like Arthur beiser and paul tipler but they just throw formulas and in example number that have to be plugged and get the answer)

@SillyGoose the integral $\int dt'\mathrm{e}^{i\omega_{kk'}t'}$ is not convergent in the ordinary sense

It's the integral representation of the Dirac delta

More formally, you would understand this by learning about Fourier transforms of distributions

@SillyGoose I guess this png is related to your Phys.SE question. Out of curiosity, which ref. is the png from?

Oh my god, this is the first time I see Qmechanic replying to a physics question in the hbar

@Mr.Feynman my heart goes !!!!!!

@peterh yeah i think this is also pointing to smth relevant ab the points i made leading to this discussion. elementary education is indeed compulsory in the US, but there are a lot of factors that make it not accurate to say that education is free imo. for instance, my fellow students and i in [...]

[...] primary school could not go to school for days and some for weeks because there were lice outbreaks and the school policy was that students must be sent home if they have lice. the living conditions were so bad that this ended up being a cycle of lice outbreaks for more than a year. [...]

[...] did this happen in the richer neighborhoods and their schools? definitely not, and in the US schools are funded partially by the property tax of the neighborhood of the school and its forbidden to go to a public school outside of your district so even then id say that its not really accurate to say even this level of education is free. [...]

[...] what does it mean to say something is free when the quality of it depends on how much you pay for it? in similar vein, china requires everyone to to go primary school (and past that ofc), and the exams to get into schools at each stage are supposed to be based on merit. however, i have multiple friends who just paid to get into schools their scores werent good enough for. imo this is also not reminiscent of "education is free" and this is a country that is entirely separate from the [...]

US. i cant imagine that germany is free of these types of issues (or really any country) since they are intrinsic to a society with any sort of class divide, but id be happy to hear if it indeed was not a problem. i just dont really believe that "education is not free" is truly a US centric statement

i think it just reflects we mean different things when we say something is free.

im curious what you think in particular about this question "what does it mean to say something is free when the quality of it depends on how much you pay for it" @ACuriousMind (tagging since it's a response to your messages in the past convo but sorry if this is not proper)

One way of constructing Calabi Yau manifolds is to consider polynomials in projective spaces. E.g. A quintic in $\mathbb{P}^4$. What I am unable to understand is if a particular quintic corresponds to a particular CY fold, and changing the coefficients of the quintic corresponds to changing parameters of the CY 3-fold i.e. we get different CY 3-folds for different coeffs, then why does the Hodge number $h^{1,1}=101$ is associated with a CY 3-fold?

Is it not counting in some sense number of different CY 3-folds of the same type...here quintics in $\mathbb{P}^4$?

what is everyone's favorite proof

@RyderRude Cantor's diagonal argument is pretty cool

trying to figure out how to word something. namely: suppose you have fluid flowing out of a large tank through a narrow tube. i want them to use Bernoulli's principle to analyze the flow within the tank and use Poiseuille's law to analyze the flow in the tube

but bernoulli isn't applicable if there's viscosity, whereas Poiseuille's law relies on it

you know it is not applicable, yet you want to use it?

tbh it's more me trying to salvage a final exam problem from last year

i'd rather not start from scratch on it but

the more i look at it the more i think i have to

i've already had to scrap part of it b/c the old solution made a BS assumption

and if you don't make said BS assumption, the problem gets unreasonably hard

Yes; and the usual assumptions are extremely bad at fitting to actual experimental data

i could just tell them to ignore viscosity for that part i guess

morally i don't like it but

Part of the issue is that the boundary layer is actually going to be a mathematical singularity. Bender & Orszag, yes that book covering some mathematical details of renormalisation, mentioned this boundary layer nonsense

yeah

viscous vs non-viscous is a headache

Of course, IRL the quantum system will not actually have a singularity, but maths will be maths and the physics will approximate what maths says it should have

viscosity is weird in that it makes things mathematically both easier and harder

harder in that you can't appeal to Bernoulli's principle

easier in that viscosity helps prevent turbulence

@SirCumference yeah.. and Cantor invented his way of talking about infinities out of the blue

@RyderRude well that's kind of how mathematics works

you establish some axioms (which are ultimately up to what you intuitively want math to be) and then derive logical statements from them

if you want to reject some of those axioms, it'd be nonstandard but still perfectly valid

@SirCumference or realize that the axioms you picked are contradictory

@Semiclassical well, you usually try to avoid that

but there's a deep rabbit hole into establishing those axioms

try being the operative word :P

in any case, there's no "best" set of axioms. it comes down to preference

Cantor was nonetheless one of the most important contributors to set theory

@SirCumference i link this image way more often than i thought when i first saw it

smbc-comics.com/comic/know-your-linguistic-philosophies

lol

@RyderRude All this is to say that there's no such thing as a fundamentally true statement in math. You need to first define what is considered true (i.e. define your set of axioms), and that's something that indeed comes out of the blue. Everything after that is just a matter of finding logical consequences of those axioms

@SirCumference yes.i meant that that idea of infinity was out of the blue. there r other inventions that r not out of the blue ,. e.g. non euclidean geometry which is an extension of euclidean geometry

well, that would be an example of a logical consequence of axioms

when you're a set theorist like Cantor, you gotta work from the beginning of it all

i meant the axioms of non euclidean geometry r not out of the blue becuz similar axioms existed before

@Relativisticcucumber China follows probably a communist model. Most important characteristics is, that the state considers important the development of its citizens, so strong education is top prio. Beside that, there is a system of "protegee". That is an un-official pyramids of strong men and their followers, similar to the feudalism. For example, if you are a friend from the University of the director of your company, then you have some chance that this friend will call his friends at the

@Relativisticcucumber University where they have influence, and they trick that your son will be accepted there, even without enough points. That is, in theory, corruption, but former socialist countries work partially still on this way. As far I know, USA works very differently, there are "clans" or "clubs", mostly organized at the Universities, and these help each other. The former commie countries have pyramides of protegees.

usually, mathematicians invent axioms to abstract out a previously known idea... like group, set, topology, etc

I don't think Cantor invented infinity :P

like, everyone knows how collections of objects work. u just codify it and u hav got a set theory

He just demonstrated how infinity works under our usual set of axioms

@SirCumference infinity was a known idea, but levels of infinity is something no one knew intuitively

u cannot find levels of infinities in historical writings i bet

well, i'm sure somebody came up with the vague notion

oh

but yes, I assume no one had formally defined it before those kinds of developments in set theory took off

maybe someone had a vague idea

the idea that a boundless continuum intuitively feels "larger" that an infinite discrete set isn't too out there

depending on how vague it was, we could or could not call Cantor's work our of the blue

@SirCumference yeah... but in that sense, rational numbers feel equally larger

but it is the vague notion, yes

indeed, there is a lot surprising once you start formalizing the concepts

the idea that $[0, 1]$ is equinumerous with all the reals is one such surprising case, at least until you read the proof

yeah

or the idea that there are "more numbers" between 0 and 1 than there are integers on the number line

yeah. i wouldve expected these two to be equal

in any case i oughta head back to QM

later y'all

bye

@Relativisticcucumber As far I know, criminal law calls this "misuse of influence". That is, for example, if the director of a University misuses his influence to trick the entry exams, to help the son of his friend to get in. But that is the theory. Practice is that everyone "knows" where is the limit and they do not pass this limit. In the West, this limit is zero. On the East, it is not.

@Relativisticcucumber Btw, I have read a German elementary school math book. Imho, if the small kids really know these, then Germany has a strong math education. But I think it is more likely, that the kids learn only a small part of that.

math physics

@ACuriousMind I just came across your answer to this question and a note on QFT which discusses this but I haven't seen this discussed elsewhere. So I just wanted to ask if you thought and wrote about it or have you heard about this argument in some class or read it somewhere if you remember?

If it is you who thought deeply about it and cooked it up then it might be possible that the author of the notes just copied stuff from your answer...otherwise it might be that both of you got hold of that rare source which I am also after :p

The note looks like this btw

@Qmechanic this png is from my own notes i was typing up but i wrote down the original form of the CS action from Witten’s jones polynomial paper

@Mr.Feynman oh okay thanks

QMech: also thanks for your answer on my q it cleared everything up >:D

@Sanjana I don't see any argument here that would require elaboration, can you be more specific?

@ACuriousMind Yeah the argument is completely fine. I was just wondering whether you read it from a book or heard it somewhere else or just thought of it and wrote it down

@Relativisticcucumber I don't understand the question since the point of free education is that it's free - you can't "pay more" to make it better. I attended one of the best universities for physics in the country and didn't pay for it (I just had to move to Heidelberg) any more than for any other university

@Sanjana I just came up with it on my own but I am not surprised someone else did, too - it's rather obvious if you've dealt with gauge theories a while

@Relativisticcucumber but to address this: there's lots of inequality here, we're (sadly) not in a classless utopia. But the way that inequality perpetuates itself in education is subtler than poor people getting worse schools

@ACuriousMind Oh

@ACuriousMind Btw can you have a look at this question of mine?

For example: The CY 3-fold described by the Fermat quintic and some other quintic in $\mathbb{P}^4$ are different but they have same Hodge numbers, right?

@Semiclassical omg @Relativisticcucumber which one are you. i am a descriptivist

@SillyGoose descriptivist in mentality, realist in expectations

@ACuriousMind the way i'd frame it is not in terms of it being free, but it being a public good vs a private one

there are costs involved, but that's usually just a way of distracting from who decides what the relevant costs are and who should bear that burden

where did all this farmer terminology in math come from?

kernel, sheaf

fiber

@SillyGoose germ and stalks too, which falls under sheafs as well

Todd Trimble wondered about that too here: hsm.stackexchange.com/questions/11816/…

interesting...

there very much is an agricultural metaphor running through those fundamental concepts of sheaf theory

why is relativity introduced by imagining placing "rulers and clocks" everywhere in space. this seems like a poor pedagogical choice that masks something that is truly abstract as something concrete :P

man i feel like lecturing a graduate quantum mechanics course would be fun :P

actually maybe a truly fun lecturing experience would be to be able to lecture at a summer school :P

i think rules and clocks language goes back to Einstein?

hm...i feel like it's like how some qm courses emphasize states as being wave functions when a state is much more general than that :P

clocks in particular definitely were used by Einstein in his paper introducing special relativity

@SillyGoose i think that reflects how long "wave mechanics" dominated presentations of QM more generally

which made sense when spectroscopy and applications of QM to chemistry were the cutting edge

matrix mechanics may have "won" in terms of what the interpretation of QM was, but for a long time wave mechanics won in terms of what the outward face of QM was

Right it seems like a historical artifact (pun intended hehe)

yes

i do feel like we sorta overstate the extent to which wave and matrix mechanics were proven to be equivalent

formally, it's not wrong to say that

but in terms of "intuition" it's not so clear

What is wave mechanics for a finite dimensional system?

there really isn't but eh

we say that an electron, say, is a two-level system due to spin

It would be interesting to take a geometric physics course that goes through the shared or similar geometrical formalisms of say classical mechanics, quantum mechanics, and field theories

but to actually observe that, what did people do? send it through a magnet and see it deflect in space

which is to say, something that's very much about position and momentum at the end of the day

I guess so it is the position or momentum space representation of things which is cares about in scattering and so on too (at least it seems)

hence why in de Broglie-Bohm you (usually) only worry about position as having a genuine ontology

to the extent that spin is "real" in that story it's because it says something about the appropriate wavefunction, namely that it's going to be a spinor wavefunction rather than a scalar one

not because you'd insist that the spin component in a Stern-Gerlach measurement had some objective value before the measurement. dBB insists on doing that for position, but doesn't for anything else

(that's a bit overstated, particularly if you look at what Bohm himself did, but it's very much the received wisdom for philosphers who like the Bohm interpretation)

this is me following sakurai's derivation of the lippmann schwinger equation. i don't understand how from $(76)$ we can obtain the lippmann-schwinger equation? surely we cannot invert $V$ in general, so I don't really see how only with the given data we can remove this left multiplication by $V$ to obtain the L-S equation

for reference, the above is what sakurai actually says