The h Bar

@bloo fairly sure that explodes?

user218912

@0celo7 yep

8:01 PM

is it supposed to?

user218912

no.

user218912

I don't think so.

user218912

it says in both cases you should find the same thing.

user218912

oh

user218912

okay so it is supposed to blow up for the infinity limit.

user218912

8:03 PM

but it wants you to express the integral as a series

ok time for a test

with NUMBERS

3 is a number

i'm using

2,3,5,1

those are all numbers, yes

well done

HOLY SHIT MY ANSWER IS RIGHT

8:05 PM

Congratulations ocelo7!

you done it!

I GAUSSIAN ELIMINATED CORRECTLY

you passed the number test!

THE SQUARE OF THAT IS THE REFLECTION COEFFICIENT

user218912

okay I don't get this, even if I were to evaluate this integral in limiting cases how would I get a series for the answer WITH $a$ in it?

user218912

am I being dumb

8:09 PM

what is the answer?

yes, probably

for example

when $x\to 0$ we have $e^x=1+x$

user218912

the answer is supposed to be a series of the form $za^x + ...$ where z and x are random constants I need to find.

user218912

and $a$ is the same $a$ as before from the integral.

@NeuroFuzzy any idea how to simplify this mess :)

@0celo7 Nothing sticks out to me, no

user218912

@NeuroFuzzy any idea how to help me?

user218912

8:14 PM

you did QFT right?

user218912

should I series expand the exponentials @0celo7

@NeuroFuzzy My answer agrees with wiki

they have

I tried some numbers and it works

Wow, fancy.

@0celo7, congrats! Glad you figured it out. I know I wasn't too helpful, but I'm glad you got it. =)

@heather well I still have to simplify that horrible expression

then do it all again for the other energy...

but that might be easier

8:17 PM

I'd brute force it. Multiply through by $e^{ik_2\ell}$ on the top and bottom and then replace it with $\cos(\theta)+i\sin(\theta)$ - you might be able to see most of the simplifications and not write it out, but that's at least like half a page of algebra haha

i think I can do $k_2\mapsto ik_2$

Or maybe a simplification could have been made earlier in the thing.

@0celo7, yes, that is rather horrible, but you've gotten this far, which is always good.

@NeuroFuzzy no, doubtful

the Gaussian elimination was nasty

but the factors of two will cancel AND you can pull out a factor of $k_1-k_2$ to get a quadratic!

8:18 PM

I need the binomial expansion of a cube

That means you're in a dream and not a nightmare

@NeuroFuzzy whoa what

noooo heck no!

@0celo7 In the numerator there's a factor of $k_1-k_2$ that you can pull out.

oh I'm stupid

I didn't even see that

how did that happen!

not how didn't I see it

how did it get there

Then you can multiply the top and bottom by $e^{ik_2l}$ and replace the exponentials with $\cos{k_2 l}\pm\sin{k_2l}$,and expand, and surely a lot of stuff will simplify.

8:21 PM

yeah

thanks

np!

@bloo I didn't. I can do fourier transforms and gaussian integrals all day but I don't know how to interpret $a_k$ in that equation.

user218912

:(

user218912

it's defined based off the scalar field and its time derivative.

user218912

which is a circular definition i think because

user218912

we're using the definition of the scalar field in terms of the operators here.

8:26 PM

@0celo7 I'm happy to hear that :)

@Sanya I didn't have it then

I have it now

it's actually correct

tfw you don't remember Euler' formula

I think I had a stroke

so what moderator is winning

user218912

I don't think they tell you.

probably not ACM

he's the Trump of PSE

what a loser

user218912

8:31 PM

wow ACM is our friend why are you saying these things?

ACM is cool - I hope he becomes moderator

@bloo they are probably just being sarcastic/joking

that sort of thing is hard to convey via chat

I'm surprised they haven't made some sort of sarcasm emoticon

user218912

it's called the sarcasm equation

user218912

0celo7 created it 1.5 years ago.

Really? What is it?

user218912

May 14 '15 at 0:55, by 0celo7

$$\int_\mathcal{M}\operatorname{ch}\left(\bigotimes_r(-1)^rE_r\right)\frac{ \operatorname{Td}(TM^\mathbb{C})}{e(TM)}$$

8:35 PM

what are the variables?

user218912

dunno ask 0celo7

Atiyah-Singer?

@heather that's the Atiyah-Singer index theorem

Uh...googling

"In differential geometry, the Atiyah–Singer index theorem [...] states that for an elliptic differential operator on a compact manifold, the analytical index [...] is equal to the topological index [...] It includes many other theorems, such as the Riemann–Roch theorem, as special cases, and has applications in theoretical physics." - Wikipedia

I simplified wrong.

question: what does the atiyah-singer index theorem have to do with sarcasm?

8:37 PM

bloo is insane

just fyi

oh, so it isn't a sarcasm equation?

darn

Question: "A straight line is drawn through the point $A(1, 2)$ and the point $B(2, 4)$. Find $\tan \alpha$ where $\alpha$ is the acute angle that this line with the horizontal line through $A$"

user218912

@0celo7 seriously?

user218912

@heather it used to be at least, look at the messages before and after that equation in the link.

user218912

May 14 '15 at 0:56, by 0celo7

RHS of Atiyah-Singer theorem is now the sarcasm emoticon

@heather so when are we starting the calculus lessons

I'm curious what starting with topology does

8:42 PM

@0celo7, hey, now would be a good time for calculus. And what exactly is topology? Is it like manifolds and all that jazz?

no

That problem was actually from a calculus textbook I'm reading through.

@heather do you know what a vector space is?

@heather that seems like analytic geometry not calculus

user218912

@0celo7 it is probably the pre-calc chapter of the book.

@0celo7, be back in a minute yes i know what a vector space is and bloo is right

user218912

8:44 PM

see I'm not insane :)

@heather Def. A topological space is a set $X$, with a collection $\mathcal T_X\subset\wp(X)$ of subsets called open sets such that (i) $X,\emptyset\in\mathcal T_X$, (ii) $\{U_\alpha\}_{\alpha\in\Lambda}\subset\mathcal T_X\implies \bigcup_\alpha U_\alpha\in\mathcal T_X$, (iii) $U_1,U_2\in\mathcal T_X\implies U_1\cap U_2\in\mathcal T_X$.

$\mathcal T_X$ is called the topology of $X$.

(The $X$ subscript can be omitted when there is no risk of confusion.)

A set $C\subset X$ is closed if $X-C=C^c$ is open.

user218912

@0celo7 wow that's so helpful for someone who doesn't know calculus.

@bloo Yeah I agree.

user218912

@0celo7 you should have had 0celo7 teach you calculus when you were a freshman/sophomore.

Why?

8:50 PM

First, before keep going into this madness, I have that mathjax thing bookmarked, but it still isn't showing up properly and I'm not sure why...

user218912

@heather you're supposed to click it.

did you drag it into the bookmark bar?

you have to activate it each time you load the page

@heather Direct initial questions at bloo. I'm still simplifying this thing lol

@bloo, thanks - it renders now.

user218912

@0celo7 simplifying what?

@bloo, in terms of questions: what in the great wide universe is 0celo7 talking about? I'm not sure what half those symbols mean.

Okay, I guess I should clarify what I do get first:

8:51 PM

@bloo reflection coefficient

user218912

@heather he's just defining a topological space.

user218912

but

@bloo, yes, but I don't know what a topological space is.

I barely understand half the symbols.

that's the point

user218912

before you can learn that you should know basic set theory to understand the symbols.

user218912

8:53 PM

@heather you shouldn't learn calculus this way.

I know the sideways U means Tx is the subset of p(X) (dunno how to format it the same) and I know the sideways epsilon/e thing means element of/member of, and that the arrow is used to map functions (whatever that means) and the upside down U means intersection, I think.

And U means union.

as far as I understand it, that's how Europeans do it @bloo

user218912

@0celo7 they know basic calculus coming from high school.

user218912

that's different.

Proof?

8:54 PM

@0celo7, okay, could you explain what I don't get here? I'm willing to give it a go your way.

user218912

@0celo7 so you're saying they don't learn calculus in high school?

@heather $\wp(X)$ is the power set

user218912

@heather he's not being serious, I think.

Well I'm not even in high school yet anyway =)

bloo is right you might want to learn some set theory

8:55 PM

@0celo7, okay, what is a power set

user218912

to be able to read that definition and understand it, you need mathematical maturity.

set of all subsets

@bloo Question, if I remove (i) does it change anything

@0celo7, ah, I see, would p(X) be a member of itself then? Because every set is a subset of itself?

Sure

Okay

user218912

8:56 PM

@0celo7 dunno, probably does because you need that to be true for it to be a topological space.

@bloo trick question!

user218912

huh?

the empty intersection is just $\emptyset$

and the empty union is just $X$

or the other way around

Okay, now I'm really confused.

@heather maybe this isn't the best way

8:57 PM

Is there a specific book I should read to understand set theory?

Jech

@0celo7, no, maybe not =)

it's PhD level but whatever

Right

Middle school $\neq$ PhD level

user218912

@heather 0celo7 is not being serious, if you want to learn calculus read "calculus early transcendentals" by stewart.

user218912

8:58 PM

I read that in grade 9.

user218912

what grade are you in?

8

(grade)

user218912

do you know pre-calc?

user218912

do you know how to solve a quadratic equation? solve trigonometric equations? or use logarithms?

solve a quadratic: yes

9:01 PM

quadratic equation is fucking hard

solve trigonometric equations: no

use logarithms: maybe/no (I have a general idea)

precalc: no

I'm in geometry this year and I've taken Algebra I

geometry is very important

And I have online access to the textbook via the school

Geometry is just about the best math.

And I'd be very much willing to work ahead

9:02 PM

Analysis is a close second

user218912

@heather then you shouldn't be doing calculus yet.

Geometric analysis is the best

user218912

your best bet is to read the appendix and chapter 1 in stewart's calculus book. @heather

user218912

and you'll have the necessary pre-calc.

@bloo, okay, so I should learn pre-calc is that what I need to do?

user218912

9:03 PM

@heather yes but pre-calc is basically a bunch of basic ideas and definitions that are based on common sense and some new math you may not know, so it shouldn't take you more than a few days to learn all of it if you're good at math.

@bloo, sweet! I think I could do it in a few days, I'm semi-decent at math.

user218912

@heather oh yeah I remembered

user218912

use paul's notes to learn pre-calc/algebra you may be missing.

@bloo, Paul's notes? Who is Paul?

user218912

tutorial.math.lamar.edu/Extras/AlgebraTrigReview/…

user218912

9:05 PM

some online university prof.

user218912

read all of those in order.

user218912

and you'll be set to learn any calculus.

user218912

except the kind 0celo7 is doing.

@bloo, =) I'll start reading

are there problems?

wtf is "pre-calc"? What is it with American math subject names that they don't make any sense to my European-educated mind? :P

9:06 PM

oh, yeah there are problems

sweet

@ACuriousMind, it is a class to prepare you for calculus.

=)

user218912

@ACuriousMind that is basically elementary algebra.

Heh, well, that's not the part that confuses me ;)

I'm confused why one doesn't name it after what it actually contains

user218912

where I live it's called math.

user218912

grade 9, 10, 11 math.

@bloo I see

9:09 PM

precalc is not algebra

it's trig

+ rotated conics

user218912

no.

@bloo shoo

user218912

it does contain algebra too.

See, not even you people can agree on what pre-calc is supposed to be! :D

It contains all the things listed

@ACuriousMind, though you do have a point

9:10 PM

I know what it is

he's just wrong

@ACuriousMind how do you treat acceleration in special relativity (SR)? You say the twin paradox needs no general relativity, but how does SR allow me to describe a turnaround motion which necessarily has acceleration? I've never been deep into relativity, but even after reading the thread you linked, it still does not agree with what I thought SR's limits are ...

user218912

@0celo7 meanie.

@Sanya Read the book by Gourhoulhon.

He has complete formulas for accelerated motion.

Spirals, accelerations, accelerating spirals

All sorts of stuff.

SR can handle acceleration just fine.

WildTrig1: Why Trig is Hard

@Sanya SR is just "GR in Minkowski space". No problem with acceleration in that.

Count Iblis

►

9:13 PM

I don't know who first propagated the myth that you can't have acceleration in SR.

user218912

@0celo7 that's what my GR prof said last year.

@bloo, okay, quick question: one of the problems is $\frac{xx^{-\frac{1}{3}}}{2x^5}$

You promptly corrected him, right?

@ACuriousMind ok, then they never taught us SR well ... that explains something at least, thanks

user218912

@0celo7 nope.

9:14 PM

@0celo7 I'll have a look

user218912

@heather huh?

and I reduced it to $\frac{1}{2x^{\frac{13}{3}}}$

@ACuriousMind While technically true, this is also misleading. One does very different things in SR vs. GR...

@Sanya Yes, SR is often taught with a strange focus on exercises with the Lorentz transformations/time dilation/length contraction and not enough focus on the invariance of the spacetime interval. As soon as you have that and know that it denotes proper time (or distance) when integrated along a worldline, you can resolve the twin paradox just fine.

but the answer given was $\frac{1}{2}x^{-\frac{13}{5}}$

9:16 PM

You know I'm still not sure how to do rotating frames in GR

but isn't it bad to keep the exponent negative as opposed to turning it into a fraction like I did? @bloo

There's this weird thing where the coordinates cease to be valid at some radius, but they never describe the coordinate patch outside

@Slereah See Straumann sect. 2.10 for the static case

I know how confused I was by my first SR lectures because all those silly "paradoxes" didn't really teach me anything except to apply the time dilation/length contraction formulae. Once I learned about the invariance of the spacetime interval, it all clicked into place.

user218912

@heather it depends, but they're the same.

user218912

9:18 PM

shouldn't it be -13/2 in the exponent?

@ACuriousMind I've really only learned to do Lorentz transformations between two frames moving with constant speed relative to each other in Electromagnetism, that's pretty much all; never had the time to do general relativity (or more special relativity, for which there is no lecture here) - but I was always offered the most abstruse "explanation" for this twin stuff that never made much sense

@0celo7 we have access to the ebook, yay~ I hope I'll find the time to read it, thanks for the title :)

@bloo, huh, I'll check over my work again

@Sanya There isn't any specific SR lecture here, either. It's a small part of about half a dozen other lectures

user218912

@heather no I'm wrong I can't read my own handwriting.

@bloo, okay

9:20 PM

WHAT THE HELL WAS I DOING WRONG FOR THE PAST HOUR

argh

this problem will kill me

@0celo7, what's wrong?

for some reason I am getting weird things

I think I multiplied by $i$ wrong somewhere

Yes

I finally got the Wikipedia answer

8 pages later

6 hours

Part a) is 1/3 done.

@0celo7, I'm afraid to ask, but how many parts are there?

3

I think this was the hard one

now I can compute the transmission coefficient by using probability conservation

then use a trick to get the other energy case

@0celo7, dunno what that means, but good luck

9:27 PM

^my day

ok, let's compute the TR coefficients.

@bloo, thanks for the paul's notes recommendation! I've learned some stuff about functions and composition already and I'm cruising through the problems I know, so all is well.

lol

gotta compute the squares of this stuff now

On to trig "review"

@ACuriousMind what's $\cos ix$?

can I write that in terms of hyperbolic trig?

indeed!

user218912

@heather good luck.

9:39 PM

@bloo, reading through the first paragraph, I was worried but okay, on the second paragraph I knew I was going to need that good luck. Darn the unit circle.

=)

user218912

@heather lol I'm not a geometric thinker either.

user218912

I dislike diagrams and visualizations.

user218912

but I like art.

@bloo, "we can see that"

for me is "magically this works"

seriously though, what do cos sin and tan (triangle relations) have to do with the unit circle?

$$T=\qty|\frac{A_3}{A_1}|^2=\frac{16k_1^2k_2^2}{(k_1-k^2)^4+(k_1+k_2)^4+2(k_1-k_‌2)^2(k_1+k_2)^2\cos(2k_2l)}$$

my god

I did it

user218912

9:42 PM

@heather it's just a visualization.

@bloo, that's the thing though...the problems ask you to solve this and that without a calculator and I don't really know how to do $\sin \frac{2\pi}{3}$ and it expects that it is easy to figure out this using the unit circle...

@heather At a point of the unit circle, you can construct a triangle by dropping the perpendicular onto the x-axis, and taking the triangle formed by it, the distance fron the point where the perpendicular meets the x-axis, and the radius from the origin to the point on the circle.

user218912

@heather just look at the solution, it should explain.

@bloo, yeah, good idea

The arc length between the x-axis and the point on the circle is the angle in radians, the length of the perpendicular is the sine of the angle, and the length of the part of the triangle parallel to the x-axis is the cosine of the angle.

9:45 PM

=)

user218912

@ACuriousMind for my integral should I expand the square first?

@0celo7, congrats!

user218912

and before you say you have no idea what i'm talking about

user218912

i pinged you

oh

I think it needs to be simplified a little more

I'll ask the prof first...

9:49 PM

Once you realize this, you can derive some special sines/cosines from ordinary triangle relationships, for instance you can figure out $\sin(2\pi/3)$ from things you might know about equilateral triangles

user218912

@ACuriousMind $$\langle 0 | \left[\int d^3x\, \bigg[ \int d^3k [a_k e^{-ik \cdot x} + a_k^\dagger e^{-ik\cdot x}]\bigg] e^{x^2/a^2}\right]^2 |0\rangle$$

user218912

please help me :(

user218912

I know that without the square it's 0 right?

@bloo Slereah already told you to solve the $x$ integral by completing the square, and you'll have to get rid of the $a_k$-crossterms by using the CCR to get them to act such on the vacua that they give zero. I'm not interested in hashing out any further details.

user218912

@ACuriousMind oh I use the CCR

9:53 PM

Obviously the question is ill defined because you can't take products of distributions

Heyoooo

@ACuriousMind I told him that days ago

user218912

wow stop making me feel bad.