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danielunderwood
danielunderwood
3:52 AM
So in Goldstein (Table 7.2 in my version), he lists $\lvert i \rangle \langle j \rvert$ as the metric for QM, but I can't seem to find anything else giving an explanation. Does that make sense?
 
 
1 hour later…
Anonymous
4:54 AM
@BalarkaSen External threats to our memedom. Get your arms and ammunition ready.
 
Anonymous
@danielunderwood Can you send a screenshot ?
 
Cup Fever
Cup Fever
why be at arms with a critical question?
 
Anonymous
@CupFever We need to fight for the survival of our kingdom ;)
 
John Rennie
John Rennie
It's an existential challenge. These threats have to be taken with the utmost gravity!
 
Cup Fever
Cup Fever
where there is mass/substance there is gravity :-)
no need to feel threatened
 
Physics Meta
Physics Meta
5:02 AM
0
Q: Tag with no description which should be merged

anna vI just discovered there is a floating ( no description) "elementary-particles" tag, which description is included in the "particle-physics" tag. Should not these be merged?

feature-request
 
Anonymous
@CupFever Just wait till you get spaghettified
 
Cup Fever
Cup Fever
mankind invented the rack centuries ago
> One powerful method for putting pressure upon prisoners was to force them to watch someone else being subjected to the rack.
:-/
 
Anonymous
5:27 AM
@CupFever Eeeeh. Sometimes ignorance is better
 
Cup Fever
Cup Fever
well, as they say; learn history because it has a way of repeating itself
 
Anonymous
@CupFever Oh, it will repeat itself regardless. Someone might even take cues from those and build a modern version of the rack, possibly a robotic one.
 
Cup Fever
Cup Fever
yeah, that's one way of looking at it
 
Anonymous
You'll always find a few humans several standard deviations away from the mean, as long as you don't genetically engineer then them to introduce uniformity. You can already shudder at the thought that the average human can be both very cruel and very kind depending on the situation.
 
Cup Fever
Cup Fever
5:45 AM
indeed, situational ethics is an entire field of psychological study
founded, of course, on philosophy
 
David Z
David Z
6:18 AM
@DanielSank ooh that's uncanny :P
 
Cup Fever
Cup Fever
niiice
i like Koz'kamun
for "obvious" reasons :P
 
Physics Meta
Physics Meta
0
Q: On-topic answers for seemingly off-topic questions

ChairI was thinking about the recent question "Why is making a white laser so difficult?" If I recall correctly, I left my usual engineering-type comment: "This may receive better answers on engineering SE." The comment has been deleted (probably by a mod), so I'm not even sure if the person who sugg...

discussion scope
 
Cup Fever
Cup Fever
HEEEELP!!! I'm being slaked by this^ meta feed :D
*stalked
 
 
1 hour later…
Secret
Secret
7:42 AM
13
Q: What is "white light" ? Uniform wavelengths or uniform frequencies ?

SomeoneSuppose you have a mixture of electromagnetic waves of wavelengths spreaded on the visible spectrum only (from $\lambda_{\text{min}} \sim 400 \, \text{nm}$ to $\lambda_{\text{max}} \sim 700 \, \text{nm}$). At some ideal detector, the light spectral distribution is described by a functional like ...

optics visible-light vision dirac-delta-distributions luminosity
Anyone have a good simulation to simulate what a white light from a uniform spectrum look like?
 
 
6 hours later…
danielunderwood
danielunderwood
2:11 PM
:45524102
 
danielunderwood
danielunderwood
2:28 PM
@Blue sorry if you get tagged multiple times. My tags keep disappearing when trying to respond to your message
 
John Rennie
John Rennie
I just heard a massive cheer from all the houses around me.
I take it that England have just scored.
 
Cup Fever
Cup Fever
perhaps
:-)
 
John Rennie
John Rennie
3:14 PM
Another cheer. Two nil?
 
Adam Uraynar
Adam Uraynar
Wow, this room is hopping compared to the EE chat room.
 
Loong
Loong
@JohnRennie yes
 
Adam Uraynar
Adam Uraynar
If I was a physics prof, I'd say something about electric permeability ;)
 
Cup Fever
Cup Fever
3:40 PM
what do you want to say about it? :P
 
Akash. B
Akash. B
guys i have some doubts
 
Cup Fever
Cup Fever
about?
 
Akash. B
Akash. B
@CupFever double slit experiment
 
Cup Fever
Cup Fever
askaway
 
Akash. B
Akash. B
@CupFever as we know in double slit experiment electrons, shows some strange movements
 
DanielSank
DanielSank
3:47 PM
@DavidZ Play MtG?
 
Akash. B
Akash. B
@CupFever does protons shows this type of movements?
 
Cup Fever
Cup Fever
do you mean "positrons"?
 
ACuriousMind
ACuriousMind
@Akash.B If by "movements" you mean the results of the double-slit, then yes. In principle, every particle behaves like this. The double-slit experiment has rather notably been carried out even with buckyballls, which are a rather large carbon molecule
 
Akash. B
Akash. B
oh I got it
 
Loong
Loong
@JohnRennie final whistle
 
Akash. B
Akash. B
3:52 PM
@ACuriousMind so what about photons?
 
ACuriousMind
ACuriousMind
@Akash.B When I say "every particle", I mean every particle.
 
Akash. B
Akash. B
okay
 
ACuriousMind
ACuriousMind
The double-slit behaviour is a general consequence of how quantum objects behave, it does not rely on any particular properties of the objects you're shooting at the slits.
 
John Rennie
John Rennie
@Loong sooo ... we're in the semi finals. Well, well, well.
 
Akash. B
Akash. B
@ACuriousMind so what is the reason for this behavior ?
 
John Rennie
John Rennie
3:55 PM
I imagine the Swedish football team will be able to get jobs at Ikea :-)
 
ACuriousMind
ACuriousMind
@Akash.B I'm sure there are plenty of explanations of the double-slit out there, for example every single textbook on quantum mechanics. It's not something that fits into one or two chat messages, at least not without oversimplifying.
 
Akash. B
Akash. B
oh i see
 
Cup Fever
Cup Fever
the Brazilian footballers don't have that luxury
 
Balarka Sen
Balarka Sen
England was very good
 
Cup Fever
Cup Fever
for a change
 
John Rennie
John Rennie
4:01 PM
@BalarkaSen I have to confess I didn't watch the game. You think it bodes well for the semi final?
 
Balarka Sen
Balarka Sen
Yes, I think the semi-final (likely England vs Croatia) would be very interesting to watch
England didn't fluke their way through the games so far.
 
Cup Fever
Cup Fever
don't count the home team out yet pal
 
Balarka Sen
Balarka Sen
lol
 
Cup Fever
Cup Fever
the refs fear the home crowd
 
Balarka Sen
Balarka Sen
maybe putin will hack his way into the semifinal
 
John Rennie
John Rennie
4:05 PM
He'll be bringing novichok along to the game.
 
Cup Fever
Cup Fever
the royal family will end their boycott too
 
John Rennie
John Rennie
Royal family boycott?
 
Cup Fever
Cup Fever
yeah, they boycotted the world cup
 
John Rennie
John Rennie
Football is too common for the royals :-)
 
Cup Fever
Cup Fever
that too
 
Loong
Loong
4:13 PM
 
Balarka Sen
Balarka Sen
the British commentators are the key attractions of the England games
they rant so much about the english glory lmao
 
Cup Fever
Cup Fever
bring football back home
that's like saying bring hockey back to Canada
 
Anonymous
@danielunderwood They seem to be using a strange notion of "metric". Afaik, the standard metric in a Hilbert space is simply the inner product $\langle j| i \rangle$.
 
Cup Fever
Cup Fever
or bring physics back to Germany :P
 
Anonymous
@ACuriousMind Could you clarify that ^ (the screenshot Daniel posted from Goldstein)
 
Anonymous
4:20 PM
 
ACuriousMind
ACuriousMind
5:08 PM
@Blue Hmmm...they seem to have botched the "vectors" column, all the differentials there should be derivatives
 
Anonymous
@ACuriousMind Ah. And what about the "Quantum theory" row? The metric there seems to be wrong
 
ACuriousMind
ACuriousMind
@Blue I don't really know what's meant by the "metric" there to begin with.
 
Anonymous
1 hour ago, by Blue
@danielunderwood They seem to be using a strange notion of "metric". Afaik, the standard metric in a Hilbert space is simply the inner product $\langle j| i \rangle$.
 
Anonymous
Yeah, same :/
 
Anonymous
It doesn't make sense
 
Anonymous
5:21 PM
But it's very rare to find mistakes in Goldstein
 
Anonymous
Which is why I was doubtful initially
 
danielunderwood
danielunderwood
5:45 PM
Yeah everything that I've found relating to the matter seems to be saying inner product is closest thing to a metric. I can't seem to find any errata related to that table either
 
Anonymous
5:55 PM
@danielunderwood Report it! :P
 
dmckee
dmckee
Actually, scratch that. It's not right.
 
ACuriousMind
ACuriousMind
@danielunderwood Sure, the inner product induces a metric (in the mathematical sense of what a metric space is) on the Hilbert space, but that's not the same as the (pseudo-)Riemannian metrics in all the other rows
 
dmckee
dmckee
The column labled "metric" in that image is showing representations of the metric(s) in particular spaces and coordinate systems.
@ACuriousMind Hmmm ... I think the author of the table would argue that they are physicist-same even though they are not mathematician-same.
Because you use can use them in a uniform way for the things that physicists care about.
 
ACuriousMind
ACuriousMind
Also, all the other (co)vectors in that table are (co)tangent vectors to a manifold, while the vectors in a Hilbert space...aren't. I'd argue the "quantum" row simply doesn't belong there - it looks similar, but it play by different rules, both physically and mathematically
 
Semiclassical
Semiclassical
6:21 PM
I think you can run into instances of Riemannian metrics in QM, but they're not the rule
e.g. the Fubini-Study metric to give distances between pure states (en.wikipedia.org/wiki/…)
 
ACuriousMind
ACuriousMind
@Semiclassical You can, e.g. the Fubini-Study metric on finite-dimensional projective Hilbert space, but that skews the analogy the table there tries to draw even further because there the states are not vectors, but points on the manifold that metric lives on
 
Balarka Sen
Balarka Sen
6:42 PM
Hilbert manifolds are the right context for infinite dimensional Riemannian manifolds
 
Anonymous
@DanielSank You might be interested in answering this.
 
Anonymous
There's a lack of experimentalists on the site :/
 
danielunderwood
danielunderwood
7:07 PM
@dmckee What exactly do you mean by physicist-same but not mathematician-same? The limited knowledge I have kind of puts the difference as $g: V \times V \to \mathbb{R}$ where $V$ is your vector space while $\lvert \cdot \rangle \langle \cdot \rvert: \mathcal{H}^* \times \mathcal{H} \to \mathbb{R}$ or is it more sophisticated than that?
I suppose it may just be something I need more knowledge to understand though
And I guess in a sense, the projection version is kind of the same in practice since we have an easy mapping between $\mathcal{H}^*$ and $\mathcal{H}$
 
ACuriousMind
ACuriousMind
@danielunderwood The metric in the non-quantum cases there is not a simple form on a vector space.There's a manifold - space, or spacetime in the case of relativity - and you have a map $g$ such as you write on each tangent space to the manifold. That's how the Euclidean spherical metric there can depend on $r$ - it's not a single $g$ on a single vector space, but it is a family of such maps, on at every point $(r,\theta,\phi)$.
That is, all the classical entries in that table are really about (co)tangent vectors to manifolds
But in the quantum case, the state vectors are not tangent vectors to anything and the tangents to the projective state space on which there is a metric do not have straightforward meaning.
 
danielunderwood
danielunderwood
So in the first part, do metrics only come associated with tangent spaces and not vector spaces in general? I know that there's also a difference between spaces with and without a metric, but haven't learned quite that far yet.
And for the second part, do we know that state vectors aren't tangent to anything or is that just taken to be the case? Although I suppose given $TM$, you could reconstruct $M$?
And is there a standard book/resource that would cover the more technical math from a physical perspective? Arnold?
 
ACuriousMind
ACuriousMind
7:26 PM
@danielunderwood The problem is that "metric" is ambiguous: You can have a standalone metric(=distance function) on a single vector space, but actually what we have here when we write down matrices or dot-products is much more than a metric, it's an inner product. On tangent spaces, it is a habit to call these inner products metrics, since they come from a true metric (=distance function) on the manifold.
And when I said that state vectors are not tangent to anything I just meant that they are not defined as tangent vectors to some manifold. It doesn't really make sense to ask of arbitrary vectors whether they are tangent vectors.
 
danielunderwood
danielunderwood
7:41 PM
Ahh that was a bit of a missing piece. I've always thought of inner products and metrics as the same. I've also thought of dot products as a type of inner product, though I'm not sure that that's the case either. I was actually watching a lecture yesterday that said the interior product is the dot product, which brings in more confusion
 
ACuriousMind
ACuriousMind
@danielunderwood Dot product and inner product are usually synonymous, yes
 
 
1 hour later…
Balarka Sen
Balarka Sen
8:53 PM
Croatia-England semifinal confirmed
 
John Duffield's dog
John Duffield's dog
how do you people like my new username
 
Balarka Sen
Balarka Sen
who are you
don't say John Duffield's dog
 
John Duffield's dog
John Duffield's dog
my previous name was too long and formed by random strings of chars
i dont even remember it
too bad I can only change name once per month
 
Anonymous
9:09 PM
@JohnDuffield'sdog ofhe_iAgDWolbuuTZO_5X1L6uuwfVP
 
Anonymous
@JohnDuffield'sdog Awful as ever ;)
 
Anonymous
But someone might be happy
 
bolbteppa
bolbteppa
9:25 PM
What a game
 
Semiclassical
Semiclassical
which
 
bolbteppa
bolbteppa
Both, but Croatia especially
 
Semiclassical
Semiclassical
is American, so ignorant of World Cup for the most part
I heard England beat Sweden tho
 
Balarka Sen
Balarka Sen
ya
 
Semiclassical
Semiclassical
which is a thing i guess?
i feel about as aware of the World Cup as the rest of the world does about American football
 
bolbteppa
bolbteppa
9:32 PM
Paying attention to the world cup is like paying attention to superbowl finals or Republican primaries, random fun even if you have no idea what's going on
 
Semiclassical
Semiclassical
(which, tbh, I'm also pretty ignorant of but that's just me)
@bolbteppa and, like in those cases, certain outcomes have the chance of leading to riots
Though quite when those riots happen depends on the particular event, of course.
Superbowl finals: right after
 
bolbteppa
bolbteppa
Yeah haha
 
Semiclassical
Semiclassical
Republican primaries: a year later
what're you doing lately re: physics
 
bolbteppa
bolbteppa
Remember when this was considered beyond the pale:
 
Semiclassical
Semiclassical
ah, for the innocent days of 2012
when we thought the world was going to end because of Mayan numerology
 
bolbteppa
bolbteppa
9:36 PM
\begin{align}
F(a,b;c:z) &= \sum_{n=0}^{\infty} \dfrac{(a)_n(b)_n}{(c)_n n!} z^n \\
&= \sum_{n=0}^{\infty} \dfrac{(a)_n}{n!} z^n \dfrac{\Gamma(c)}{\gamma(b) \Gamma(c-b)} \int_0^1 t^{n+b-1} (1-t)^{c-b-1} dt \\
&= \dfrac{\Gamma(c)}{\gamma(b) \Gamma(c-b)} \int_0^1 t^{b-1} (1-t)^{c-b-1} \sum_{n=0}^{\infty} \dfrac{(a)_n (tz)^n}{n!} dt \\
&= \dfrac{\Gamma(c)}{\gamma(b) \Gamma(c-b)} \int_0^1 t^{b-1} (1-t)^{c-b-1} \dfrac{1}{(1 - tz)^a} dt \\
&= \dfrac{\Gamma(c)}{\gamma(b) \Gamma(c-b)} \int_0^1 t^{b-1} (1-t)^{c-b-1} (1 - tz)^{-a} dt
 
Semiclassical
Semiclassical
oh hey, hypergeometric
 
bolbteppa
bolbteppa
A Course of Modern Analysis
A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by E. T. Whittaker and G. N. Watson, first published by Cambridge University Press in 1902. (The first edition was Whittaker's alone; it was in later editions with Watson that this book is best known.) Its first, second, third, and the fourth, last edition were published in 1902, 1915, 1920, and 1927, respectively. Since then...
@BalarkaSen you would learn more reading that for a year than anything else in your course
 
Semiclassical
Semiclassical
Here's something 'fun' with hypergeometric functions. it's probably not shocking that, if you take various limits of them (e.g. with parameters coalescing and appropriate changes of variables) you can get to other known special functions
 
bolbteppa
bolbteppa
This stuff is actually bananas
All those limits you take give you things which actually aren't hypergeometric or even Fuchsian
But you can still apply those methods because you just moved the singularities to coincide
 
Balarka Sen
Balarka Sen
Classical function theory
Good stuff
 
bolbteppa
bolbteppa
9:39 PM
It has actually taken me years to make sense of basic things in this ****ing book
 
Semiclassical
Semiclassical
with the diagram given here giving the appropriate degenerations: en.wikipedia.org/wiki/Painlev%C3%A9_transcendents#Degenerations
But what you should notice is that the article I'm linking to is about the Painleve transcendents, i.e. solutions to certain nonlinear ODEs
 
bolbteppa
bolbteppa
The Hamiltonian systems thing below that
Absolutely terrifying
Trying to make sense of it on a time limit
 
Semiclassical
Semiclassical
lol
painleve is scary af
 
bolbteppa
bolbteppa
Of course got nowhere
There's a chapter on them in here
 
Semiclassical
Semiclassical
oh hey ince
 
bolbteppa
bolbteppa
9:42 PM
Skim it randomly and throw it away promptly
 
Semiclassical
Semiclassical
"differential equations in the complex domain" nice
isomonodromy theory is cray cray
(translation: any math i don't understand is cray cray)
 
bolbteppa
bolbteppa
The coolest thing I found in a different book was determining when a linear second order ode reduces to a constant coefficient equation by a change of variables
So you can actually prove those Euler-Cauchy equations reduce to constant coefficient equations not just pull it out of thin air
 
Semiclassical
Semiclassical
It of course matters how generic of a change of variables is allowed
the usual one I think in terms of is $z=au^p$
though that's not really generic enough for something like Riemann's ODE
 
bolbteppa
bolbteppa
'The Cauchy-Euler equation
\begin{align}
0 &= ax^2 y'' + bx y' + cy \\
&= y'' + \dfrac{b}{ax}y' + \dfrac{c}{ax^2}y = y'' + P(x)y' + Q(x)y
\end{align}
has a constant coefficient invariant of the form
\begin{align}
J &= \dfrac{Q' + 2 P Q}{Q^{3/2}} \\
&= \dfrac{-\dfrac{2c}{ax^3} + 2 \dfrac{b}{ax} \dfrac{c}{ax^2} }{(\dfrac{c}{ax^2})^{3/2}} = \dfrac{-\dfrac{2c}{ax^3} + 2 \dfrac{bc}{a^2x^3} }{(\dfrac{c^{3/2}}{a^{3/2}x^3})} = a^{3/2} \dfrac{-\dfrac{2c}{a} + 2 \dfrac{bc}{a^2} }{c^{3/2}}
\end{align}
which is independent of $x$. This implies that the equation is reducible to an equation with constant…
 
Semiclassical
Semiclassical
sounds right
 
bolbteppa
bolbteppa
9:47 PM
Obviously need to derive where $J$ comes from, but the fact this exists is incredible
 
Semiclassical
Semiclassical
there's a modicum of a connection to what I'm saying in that the last equation is $e^x=t'^{\lambda}$
but only a modicum
 
bolbteppa
bolbteppa
Cauchy–Euler equation
In mathematics, a Cauchy-Euler equation (most known as the Euler-Cauchy equation, or simply Euler's equation) is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of the particularly simple equidimensional structure the equation can be replaced with an equivalent equation with constant coefficients which can then be solved explicitly. == The equation == Let y(n)(x) be the nth derivative of the unknown function y(x). Then a Cauchy–Euler equation of order n has the form ...
I must have read 50 intro books just throwing these things at you out of nowhere
 
Semiclassical
Semiclassical
what makes that less surprising to me is that, when I see something like $Ly=0$ with $L=ax^2 D_x^2+b x D_x+c$
 
bolbteppa
bolbteppa
Yeah it looks like a quadratic
Makes perfect sense to just plug in $x^a$ or whatever
 
Semiclassical
Semiclassical
my reaction is immediately to let $x=e^u$ so that $x D_x=D_u$
 
bolbteppa
bolbteppa
9:50 PM
Ah
 
Semiclassical
Semiclassical
and then $(x D_x)^2 = x^2 D_x^2+x D_x$
 
bolbteppa
bolbteppa
Yeah that's another way of dealing with these
Can even deal with hypergeometric with that thinking
 
Semiclassical
Semiclassical
which is more conveniently written as $x^2 D_x^2 = xD_x (x D_x-1)$
I think the general formula is something like $x^n D_x^n = xD_x(xD_x-1)\cdots (x D_x-n+1)$
which in turn looks like the sort of factorials you get with the hypergeometric series
anyways. That immediately means that the differential operator can be written as $L=a D_u(D_u-1)+b D_u+c = a D_u^2+(b-a)D_u+c$
and huzzah, you've got a constant coefficient ODE
 
bolbteppa
bolbteppa
Right now my way of thinking is Riemann equation as equation with 3 regular singularities -> Hypergeometric as canonical form -> Associated Legendre as main example -> Confluent Hypergeometric as case where two coincide -> Associated Laguerre as main example (polynomial solutions) -> Whittaker as canonical confluent hypergeometric -> radial schrodinger hydrogen as canonical example -> Hermite and Bessel as confluent hypergeometric after changes of variables I still need to make obvious
It has taken so much work to see such crazy equations as natural and fitting into a story I can't believe even exists
 
Semiclassical
Semiclassical
well, that degeneration picture I gave earlier suggests that you need to allow two possible directions in the middle
namely that Kummer's equation can degenerate into either Bessel or into Hermite-Weber
 
bolbteppa
bolbteppa
9:56 PM
Yeah what I have found looks pretty insane, but writing this up takes absolutely ages, even things like beta functions I forget and need to fit into this picture
 
Semiclassical
Semiclassical
The really horrifying thing is that, on the nonlinear side, this all somehow gets encapsulated into Riemann-Hilbert stuff
 
bolbteppa
bolbteppa
I found a book that changes variables in Kummer and gets Bessel and Hermite as different choices in the result of that, but its gigantic and can't be that complicated
I tried to learn that stuff, tried the Pain transcendents which were a pain, got scattered info, hoping to systematize it
It's really cool if crazy
"can be put into one of fifty canonical forms"
"forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them"
 
Semiclassical
Semiclassical
heh, pain transcendents
 
bolbteppa
bolbteppa
Even simple things are immensely confusing, like why you care to discuss gamma functions here - only today did it click that you want to because you can express those Pochammer's in terms of Gamma's and then analytically continue your series by just re-expressing the Poch's, it's not just because they arise by trivial re-writing, there's a necessity for them
Someone latexed up half of W&W
 
Semiclassical
Semiclassical
somehow it's all to do with monodromy
on that note, you might dig these notes: pages.uoregon.edu/njp/beukers.pdf
 
bolbteppa
bolbteppa
10:08 PM
"In general, one can go from the differential equation to its monodromy group, although it’s not very easy. The inverse problem, which for some reason is another thing called the Riemann–Hilbert problem, is much harder: given any group of matrices with a relation (...) , it is possible to find a differential equation with the group generated by M" people.ds.cam.ac.uk/rc476/furthercomplexmethods/…
 
Semiclassical
Semiclassical
yeah, they talk about that specifically in those notes
 
bolbteppa
bolbteppa
Cool
I think this is why monodromy arises: "Differential equations discuss local behaviour of functions, but their solutions are somehow meant to be global objects: when the differential equation has regular singular points, solutions to the differential equation in general have branch points at the regular singular points, and hence are not quite global objects on the Riemann sphere."
Forget exactly why they have branches
 
 
1 hour later…
Physics Meta
Physics Meta
11:33 PM
0
Q: Why won't you spend your reputation (on bounties)?

Steven SagonaIs it because your total reputation is in front of your name and the size of that number is a sort of prestige? In another discussion when someone expressed frustration that over reputation gain being bias in favor of answering simple questions, it was answered: I think it is correct that t...

discussion feature-request
 

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