The h Bar

Emilio Pisanty

7:00 PM

superior to anything I would qualify as passable? No.

Novalium Company

@Blue Isn't time pointless in the quantum world. Aren't things moving too fast and weird stuff happen with relativity?

Anonymous

@NovaliumCompany No.

ACuriousMind

@NovaliumCompany There are cases in which you want to look at a wavefunction that depends on time and cases where you don't. If your book doesn't explain that, I would submit it's not a good book :P

Semiclassical

I'm actually not remembering the exact consequence of regular vs. regular singular vs. irregular singular point

Anonymous

You don't have to worry about relativity, now.

Emilio Pisanty

7:01 PM

@NovaliumCompany In some situations the wavefunction depends on time. In other situations it doesn't. The notation adapts to the requirements of the situation.

ACuriousMind

@NovaliumCompany Standard quantum mechanics has nothing to do with relativity - and time is not "pointless" even in relativity!

Anonymous

In case the wave function changes with time, it is a function of both $x,t$. Then you write it as $\psi(x,t)$, otherwise just write $\psi(t)$

Balarka Sen

Wikipedia says $y'' + a(x)y' + b(x)y = 0$ has an regular singular point at $x = x_0$ if $a(x)$ has a pole of order <= 1 at $x = x_0$ and $b(x)$ has a pole of order <= 2 at $x = x_0$.

0celo7

@ACuriousMind time is just the fourth dimension, maaaan

Semiclassical

for second-order ODEs, I think regular singular means you can write down Frobenius solutions in the vicinity of that point

Balarka Sen

7:02 PM

If something is not a regular singular point it's irregular

Novalium Company

Ok then. Thanks. Last thing. Ψ(x, t) must be equal to something right? To what? I mean, since it's a function Ψ(x, t) = ?

Semiclassical

so having irregular singular points means solutions don't even have to be that nice

Emilio Pisanty

@NovaliumCompany Your question is equivalent to saying "I sometimes see x used for position, e.g. when calculating the static equilibrium points of a system, and I sometimes see x(t) used with a time dependence, for describing trajectories"

there's no contradiction - just different situations.

Balarka Sen

Now I don't think I understand why that's equivalent to essential singularity of the solution at $x = x_0$

Emilio Pisanty

@NovaliumCompany Nope, there's plenty of time-dependent phenomena in QM.

Semiclassical

7:03 PM

@BalarkaSen I don't think that's the precise statement, no

Novalium Company

Ok I got it, time is important :D

Emilio Pisanty

56

A: Is there oscillating charge in a hydrogen atom?

In this specific instance you are correct. If you have a hydrogen atom that is completely isolated from the environment, and which has been prepared in a pure quantum state given by a superposition of the $1s$ and $2p$ states, then yes, the charge density of the electron (defined as the electron ...

here's one example

Anonymous

@NovaliumCompany It could be equal to any function of $x,t$.

Semiclassical

@BalarkaSen See for instance this: en.wikipedia.org/wiki/…

Anonymous

You need to define it according to the situation

Emilio Pisanty

7:04 PM

amazon.com/…

↑ there's a huge slew of examples

hmmm. didn't Amazon links onebox?

Novalium Company

@Blue I don't get it. I mean, f(x) = for example x + 2. Ψ(x, t) = ?

Balarka Sen

@Semiclassical So what's the geometry/dynamics at irregular singular points?

Semiclassical

confusing, that's what it is

Balarka Sen

You're supposed to know that, physicist!!!

Emilio Pisanty

@NovaliumCompany depends on the situation.

Novalium Company

7:05 PM

Ok please give an example.

0celo7

@BalarkaSen are physicists supposed to know that?

Anonymous

@NovaliumCompany Say Ψ(x, t) = xt or Ψ(x, t) = x+t, etc

Balarka Sen

This is why I hate you people

0celo7

@ACuriousMind do you know anything about ODEs?

Novalium Company

Oh, I got it.

Balarka Sen

7:05 PM

I have nothing but haaaaate for you

0celo7

@BalarkaSen I am announcing my intention to flag.

10 seconds

5 seconds

ACuriousMind

@0celo7 Don't.

Semiclassical

"Otherwise the point a is an irregular singularity. In that case the monodromy group relating solutions by analytic continuation has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions."

0celo7

@ACuriousMind Don't what?

I thought we were supposed to use our words before flagging?

Semiclassical

ugh

don't give the rest of us a headache

Emilio Pisanty

7:07 PM

@Semiclassical you lost me at "monodromy group"

Semiclassical

lol

ACuriousMind

@0celo7 Yes, but you're not supposed to say "I'm gonna flag that" but instead explain what bothers you about the specific message/situation.

Emilio Pisanty

it's the kind of term that makes me go "well, shit. I should know this. But I definitely don't."

0celo7

@BalarkaSen Hate isn't cool man

Semiclassical

basically, how the various solutions of an ODE transform under analytic continuation

Balarka Sen

7:07 PM

@0celo7 HAH

Emilio Pisanty

@Semiclassical hmmmmm

Balarka Sen

Great use of words there, Shakespeare

Fucking Dante

Emilio Pisanty

@BalarkaSen ah

ugh

Semiclassical

i.e. if you analytically continue one of the two solutions of a second-order ODE around a regular singular point, you'll end up with a linear combination of the two solutions

Emilio Pisanty

pffff

Novalium Company

7:08 PM

@Blue Let's say I want to calculate (In a double slit experiment with electrons) the probability of an electron landing on x = 2m, without using time. How am I going to do that. I know I don't understand the stuff yet, but I'm still learning.

0celo7

@BalarkaSen ok, please no foul language

Semiclassical

and the monodromy group contains all the info about that

Emilio Pisanty

@Semiclassical oh, and the monodromy group is the matrix encoded by that linear combination?

Balarka Sen

@EmilioPisanty I only know about the monodromy group in the context of Riemann surfaces, which I can explain if you want.

Actually maybe you know that already

Semiclassical

if there's only one singular point, yeah

Emilio Pisanty

7:09 PM

@Semiclassical ah

Semiclassical

a lot of ODEs have more than one regular singular point, though

Emilio Pisanty

@BalarkaSen that's probably more intense than I have mental bandwidth for at the moment

Semiclassical

in which case each regular singular point generates such a matrix

This does connect with the Riemann surface stuff insofar as a lot of the solutions to differential equations in mathematical physics can be understood as integrals on Riemann surfaces.

Airy's integral isn't one of them, though...wait. hmmm

Emilio Pisanty

btw, whatcha make of this?

> Unfortunately we couldn't access the thesis via the link you sent us, as it seems to be located on a private server, and so it is difficult to check this on your behalf. We would appreciate it if you could send us a copy of the relevant pages (p5?) of the thesis that you mentioned.

...

if you couldn't access my thesis

Anonymous

@NovaliumCompany I think you should learn about probability distributions first

Emilio Pisanty

7:11 PM

then how did you figure out which page the copyright declaration is on?

Semiclassical

huuuuh

Balarka Sen

@Semiclassical Ah

@EmilioPisanty Hah fair

Anonymous

More importantly probability density functions

Novalium Company

Where, for free :D?

Balarka Sen

@0celo7 Fuck, I'm sorry

Semiclassical

7:12 PM

iirc you can view Airy's ODE as the confluence of two regular singular points in the hypergeometric ODE?

Balarka Sen

I'll try not to do that again

Semiclassical

and confluence of singular points makes things weird

Emilio Pisanty

@Semiclassical ugh

Anonymous

@NovaliumCompany Check Khan Academy

0celo7

@ACuriousMind Ok, I asked Balarka not to use foul language and he is deliberately trolling me.

Emilio Pisanty

7:13 PM

@Semiclassical but then that's true of, like, 50% of special functions, no?

Novalium Company

@Blue I just clicked on the video, and you said that :D

Semiclassical

yeah, for instance Bessel functions are like that

Balarka Sen

@0celo7 Oops, I fucked that up didn't I

Emilio Pisanty

@Semiclassical unsurprising

Semiclassical

The biiiiig exception to all of this are the Painleve transcendents

Emilio Pisanty

7:14 PM

@Semiclassical possibly also the Gamma function?

Semiclassical

But once you do that you're doing nonlinear ODE and then you're in for a world of hurt

0celo7

@EmilioPisanty the amazing thing is that the spacing is right, so they did log\, u instead of \log u. Incredible.

Semiclassical

@EmilioPisanty Maybe

ACuriousMind

@0celo7 @BalarkaSen I'm really not in the mood for this, can we just drop this?

Emilio Pisanty

which ones did Berry say?

Semiclassical

7:15 PM

Not sure. I know the Gamma function has Stokes lines

Balarka Sen

@ACuriousMind Ok, sorry.

Semiclassical

but I'm not sure how it works vis a vis differential equations

Emilio Pisanty

> For example, Mathieu functions fall outside the hypergeometric class, and gamma and zeta functions are not the solutions of simple differential equations

Semiclassical

Mathieu functions are awful

Emilio Pisanty

(Why are special functions special?)

Semiclassical

7:16 PM

What I always really wanted to understand were finite-gap potentials. (by comparison, the mathieu equation corresponds to a cosine potential with infinitely-many gaps)

but I could not understand it

the Inverse Scattering Transform is tough going

Emilio Pisanty

@Semiclassical well, that one is worth learning

Semiclassical

Yeah

Emilio Pisanty

@Semiclassical wait, what?

Semiclassical

yuuuup

Emilio Pisanty

whatcha mean by finite-gap vs infinitely-many gaps?

Semiclassical

7:18 PM

lemme find a link

It's in the sense of band theory.

ACuriousMind

I have heard the CM theorists whisper about gapped potentials, but I've not idea what they actually are

Bernardo Meurer

@ACuriousMind Working in COBOL clearly made you lose your soul

Balarka Sen

LOL

ACuriousMind

@BernardoMeurer I'm flattered you think that I ever had one :P

Balarka Sen

who even uses COBOL in this day and age

Bernardo Meurer

7:23 PM

I'm a good Christian man, of course I think you had a soul

@BalarkaSen ACM

Banks

Old people

Balarka Sen

Fucking hell

0celo7

This is a Christian server

no soulless people allowed

Bernardo Meurer

True

NOT ON MY CHRISTIAN CHAT

(g)old

0celo7

Far Cry 5 is about misunderstood Christians you know

Balarka Sen

I only believe in Pagan Christianity

Semiclassical

7:24 PM

systems for which the idea of upgrading your OS is a terrifying suggestion

Balarka Sen

I have donated my soul to the pagan gods

Emilio Pisanty

@BernardoMeurer ah, but are you John Oliver's #NotMyChristian?

Bernardo Meurer

@Semiclassical Ubuntu

@EmilioPisanty My name is not Christian, no

Emilio Pisanty

u sure bro

?

=P

Balarka Sen

Christian Meurer is his real name

I confirm this

0celo7

7:25 PM

@BernardoMeurer hey do you wanna talk. I've got to fill out this Brazilian visa application

Bernardo Meurer

::checks passport::

I'm positive

@0celo7 Sure, gimme a call

Balarka Sen

@0celo7 r u going to the ICM

0celo7

how do I apply for a weapons permit

@BalarkaSen yes

going to get shit-faced with Yau

Semiclassical

bleh, I can't remember where I learned what I did about finite-gap potentials

mutter

0celo7

well, hopefully I can at least meet him

Bernardo Meurer

7:26 PM

@0celo7 You can't

I can though

0celo7

@BernardoMeurer I don't trust a Brazilian near me with a gun

no offense

but you're all kind of shady

Bernardo Meurer

Semiclassical

This long article is a good summary of the history: pdfs.semanticscholar.org/a442/…

Bernardo Meurer

Don't worry, our police has got your back

0celo7

@BernardoMeurer is this 805 number yours?

Bernardo Meurer

7:27 PM

Yea

Semiclassical

But I'm not finding an easy summary of finite-gap integration

Bernardo Meurer

805-886

0celo7

I didn't know I actually had a number for you. Huh.

I've always just facetime audio'd

Semiclassical

I guess the simplest example is just the Lame equation?

Balarka Sen

@Semiclassical lame

0celo7

7:28 PM

@Semiclassical nah, that's lame

Semiclassical

lol

Bernardo Meurer

@Semiclassical Wait, is it really called lame equation?

Semiclassical

Lamé function

In mathematics, a Lamé function, or ellipsoidal harmonic function, is a solution of Lamé's equation, a second-order ordinary differential equation. It was introduced in the paper (Gabriel Lamé 1837). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates. In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials. == The Lamé equation == Lamé's equation is d 2 ...

0celo7

@Semiclassical imagine being named Dr. Lame Tits

2

Bernardo Meurer

LOL

Semiclassical

7:29 PM

no, I was just being Lazy

Balarka Sen

@0celo7 Oh god.

Semiclassical

well

Cox–Zucker machine

The Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines if a given set of sections provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface E → S where S is isomorphic to the projective line. The algorithm was first published in the 1979 paper "Intersection numbers of sections of elliptic surfaces" by Cox and Zucker and it was later named the "Cox–Zucker machine" by Charles Schwartz in 1984. == References == Cox, D. A.; Zucker, S. (1979). "Intersection numbers of sections of elliptic surfaces". Invent. Math. 53: 1–44...

Emilio Pisanty

@0celo7 Relevant context for any potential flaggers: Lamé, Tits

Semiclassical

from the talk page: "It's real, all right - Mr. Cox is my father, and he wrote the paper with Mr. Zucker specifically because it would be the "Cox-Zucker" paper"

ACuriousMind

@EmilioPisanty Risky move there, trusting yourself to find the links within the edit window ;)

Semiclassical

7:31 PM

"Risky move there, trusting people judging spam flags to consider context"

Balarka Sen

There's a quora post where someone asks if Jacques Tits' last name is really pronounced like an american would pronounce tits and right below there's an answer by Jacques Tits' grandniece

Semiclassical

(my cynical take)

Balarka Sen

a-w-k-w-a-r-d

Emilio Pisanty

@ACuriousMind I believe in living dangerously =P

@Semiclassical that was the real danger, indeed

0celo7

@BernardoMeurer brazil.travisa.com

Bernardo Meurer

7:34 PM

@0celo7 travel.state.gov/content/travel/en/international-travel/…

Emilio Pisanty

@BalarkaSen maybe we should add a clarifier to the room description?

ACuriousMind

@EmilioPisanty I'm not sure that "semi-regular discussion of Tits" is a good tagline...

0celo7

What if Tits marries Cox but does the double name

Balarka Sen

Lol

@0celo7 Oh my god stop

Emilio Pisanty

@Semiclassical It has happened before

at least on occasion

Semiclassical

7:36 PM

was not doubting you

I am pretty cynical about how flagging works

both w/r/t how it's applied and how it's assessed

Emilio Pisanty

@BalarkaSen 0celo7 is American. If I understand correctly, you need to say "omaigawdstaaahp".

Balarka Sen

Well unless he's an anime boi in which case,

Oh my gah!

►

0celo7

I'm German, thank you very much

Emilio Pisanty

@0celo7 ah

then more of a "ach, mein Gott, bitte halt!"?

Semiclassical

Nichijou 05 - GOD IS DEAD

►

ACuriousMind

7:40 PM

...how did we get here from a discussion of gapped potentials?

Semiclassical

I guess this is more appropriate: youtube.com/watch?v=P-3GOo_nWoc

@ACuriousMind eh, I sorta gave up on that

ACuriousMind

@EmilioPisanty That doesn't sound very idiomatic to me, actually :P

Emilio Pisanty

@ACuriousMind we were playing snakes and ladders and we hit a snake labelled "The h Bar"?

ACuriousMind

@EmilioPisanty Sounds legit

Emilio Pisanty

I'm pretty sure there's always a 10- to 15% chance that any discussion of stuff associated with Gabriel Lamé on this chatroom will send the conversation careening into a discussion of the names that just appeared.

Semiclassical

7:43 PM

oh, right, Lamé equation

main point with that is that the continuous spectrum has one gap

Emilio Pisanty

I don't know that there's ever been a legit discussion of Jacques Tits's work, let alone one that wasn't immediately followed by such a careening trajectory.

Semiclassical

whereas the Mathieu equation has infinitely many

Emilio Pisanty

@Semiclassical what?

ain't no gaps there

no?

Semiclassical

in the spectrum, there are

Emilio Pisanty

@Semiclassical between what and what?

this is $H=\frac12 p^2$ over $\mathbb R$ we're discussing, right?

Semiclassical

7:46 PM

No.

Emilio Pisanty

ah

Semiclassical

Mathieu is $H=\frac12 p^2+\cos \theta$

Lame still has a periodic potential but it's an elliptic function

Emilio Pisanty

@Semiclassical and that's the one with no gaps?

Semiclassical

actually, it has one gap. but just one

Balarka Sen

@Semiclassical Jesus christ

Semiclassical

7:49 PM

one way to look at it: you start with $H_0=p^2/2$, which has all periodic and antiperiodic solutions (except the singly-degenerate ground state $\psi=1$) being doubly-degenerate

Emilio Pisanty

@Semiclassical wait

domain?

boundary conditions?

Semiclassical

If you're thinking in terms of periodic/antiperiodic, it's $[0,2\pi)$

but antiperiodic solutions are 4pi-periodic on the real line

This is all band theory stuff.

You're interested in solutions which remain finite on the real line

Emilio Pisanty

huh

ok

Semiclassical

Since the potential is periodic, that means you end up with solutiosn of the form $\psi(x)=\phi(x)e^{i k x}$ with $\phi(x)$ being periodic

and real $k$ (if it's not real, then the solution diverges as x goes to plus or minus infinity)

Emilio Pisanty

yes

Semiclassical

7:54 PM

and that leads to pictures like this:

encrypted-tbn0.gstatic.com/…

where the dashed line is what you'd have if the potential was zero

and the potential makes those into avoided crossings. (they've got $\kappa$ instead of $k$ there)

the point, though, is that in the cosine potential case you turn all crossings into avoided crossings

and wherever that happens, you get a gap

Emilio Pisanty

@Semiclassical yes

Semiclassical

with the implication being that the solutions to the Schrodinger equation at that particular energy have $k$ being complex

Emilio Pisanty

and with Lamé that only happens with one of the gaps?

Semiclassical

Right.

Emilio Pisanty

ah, fair enough

that's a pretty special case, though

Semiclassical

7:58 PM

yeah

you can also do cases with finitely-many gaps not closed

Emilio Pisanty

42 mins ago, by Semiclassical

What I always really wanted to understand were finite-gap potentials. (by comparison, the mathieu equation corresponds to a cosine potential with infinitely-many gaps)

I was confused by "finite gap" implying that the gap's size was finite

Semiclassical

ah, yeah

Emilio Pisanty

-1

Q: Proving that $⟨\phi_{n} \mid a + b \mid\phi_{n} ⟩ = ⟨ \phi_{n} \mid a \mid\phi_{n} ⟩ + ⟨ \phi_{n} \mid b \mid\phi_{n} ⟩ $

I am trying to prove to myself that this relation is true. $$⟨\phi_{n} \mid a + b \mid\phi_{n} ⟩ = ⟨ \phi_{n} \mid a \mid\phi_{n} ⟩ + ⟨ \phi_{n} \mid b \mid\phi_{n} ⟩ $$ Where $a$ and $b$ are 2 operators. So the way I was trying to solve this was by writing the state vector $\mid\phi_{n} ⟩$ as...

quantum-mechanics

Semiclassical

hence why i said 'infinitely-many gaps' for Mathieu, though

Emilio Pisanty

poor guy

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