The h Bar

Physics Guy

12:02 AM

knzhou

hey!

Physics Guy

Hey !

knzhou

The atmosphere in this room has really changed in the past few months...

It's gone from mostly technical physics discussion with ACM, to hating on ACM and talking about 'math-free' physical theories.

4

Physics Guy

Yeah. That could be. i dunno it because I'm just here since June.

ACuriousMind

At least I'm not imagining that, what a relief :P

knzhou

12:05 AM

Like, if you make a connectivity graph of who talks with whom, there are clearly two completely separate components.

Except with attacks from one component on ACM in the other. :P

Physics Guy

Tell me about those good ol' days

knzhou

Well, I'd come in, read about the latest problem 0celo7 is working on, and leave. Much more pleasant!

Physics Guy

@knzhou I wont say who is the generator of this theme-changing here in h bar. We all know the answer.

But Ocelo7 still talks about his (mathematical) ideas and problems. Leastways he does.

vzn

12:22 AM

@Mast thx for your attn! & what was your vote? & maybe you prefer the 2 paragraph 69v top answer with no citations. so go vote for that one instead. (its definitely much easier/ quicker to understand! if one can say that its actually an understandable explanation...)

> "What is a photon?" we need to be honest and say that we don't really know.

Mast, Netherlands

107 6

Sir Cumference

Does anyone think it'd be possible to become a doctor while also learning GR and QFT?

vzn

← now taking physics advice from some with less than ½ my own rep :P

ACuriousMind

12:38 AM

@SirCumference If you like neither sleep nor a social life, possibly :P

bolbteppa

@SirCumference Yeah, just go through these lectures theoreticalminimum.com/courses and/also these youtube.com/channel/UCLjdjfV5J-xsb2KP4AO-bZw/playlists as you study and forget about society ;)

Sir Cumference

Sigh... ;-;

bolbteppa

@0celo7 did you get the sine derivatives

vzn

@ACuriousMind lol you say that like its a bad thing :P

Sir Cumference

I know being an astrophysicist makes no money, but I also really want to learn everything

DanielSank

12:39 AM

'sup y'all?

bolbteppa

Those theoreticalminimum videos are for someone like you dude, no reason not to try them out, has books on the first two courses out already and a third coming soon

Sir Cumference

Still tho...

I also don't really wanna be crushed by the stress

There really is no right solution...

bolbteppa

Getting past med school ultimately requires memorizing 3 books, the First aid books, not expected to think don't worry, save that for physics ;)

Sir Cumference

I guess I could keep majoring in physics and apply to med school, right?

Would that give me enough time to become fluent in GR and QFT?

bolbteppa

Are you in your undergrad?

Sir Cumference

12:46 AM

Yep

ACuriousMind

@SirCumference Walk before you can run. Are you perfectly comfortable with SR and standard QM?

Sir Cumference

@ACuriousMind As of now? Not at all.

bolbteppa

I would say do your med school stuff as best you can and don't let physics distract you, use those theoreticalminimum videos if you haven't done the groundwork courses

If you actually sat through those theoreticalminimum videos zero, one or two a day for the next six months and did them all, or most of them, you'd get enough of a feel to know whether you want to re-do every one of those mini-courses in more detail and go through all the problems, all the technicalities etc... or not, those videos are made for people who can't put in the time of wading through the pedantic details but want to get the real jist of what's going on

They wont give you the best way to think about these things, but they will give you the in-depth jist to have a real feel for what's going on, which is most likely what you're looking for

Learning QM well enough to learn statistical physics well enough to learn chemistry properly to learn biochemistry properly would be well worth the effort though

0celo7

@SirCumference fluent in GR doesn't make sense

It's a huge field

@bolbteppa nope

Too advanced

bolbteppa

What's wrong with www-math.mit.edu/~djk/18_01/chapter05/proof02.html

If you can't remember $\sin(x + h) = \sin(x) \cos(h) + \sin(h) \cos(x)$ then use Euler's formula to prove it, there is a proof and visual explanation of the meaning of it here betterexplained.com/articles/…

If those two limits are a problem, proofs here en.wikipedia.org/wiki/…

If you want to cheat, define sin and cos as solutions of a 2nd order ODE, solve it using power series to define sin and cos using power series, prove the above identity using Euler's formula (or your power series expansions), then you can treat the limit proof as an exercise, but proving those two tough limits then becomes easy because you can use the series expansion to verify them

It's not a cheat, but it's not as nice := cheat

0celo7

1:11 AM

@bolbteppa Euler's formula is proved using Taylor series

no cheating

Sir Cumference

@0celo7 True dat

@0celo7 All right, knowledgeable enough in GR that I can hold lengthy, in depth conversations about it with people here

Or regularly answer the questions about GR on the site

0celo7

@bolbteppa tbh I want to see trig explained from ZFC

bolbteppa

I did too, the best thing I've come up with so far is to use the integral definition of arcsin as your starting point

I've found 2 books doing it this way

0celo7

Hmm, $\int\sqrt{1-x^2}\mathrm dx$?

I don't know calculus

bolbteppa

If you wanted to define trigonometric functions like $\sin(\theta)$ you need to know what $\theta$ is first, how it relates to a circle, what $\theta = \pi$ is, and how this number really relates to a circle, and you need to do it in a way that links to abstract mathematics not 2-D Euclidean geometry where you're using pictures and not really sure how to justify things right?

You know that $\ln(x) = \int_1^x \frac{1}{t}dt$, i.e. the stupid Logarithm is defined using integrals in calculus books and you can take that as your starting point with this function

Why not do the same thing with trig functions

0celo7

1:24 AM

so what's the integral

I'm not sure what's wrong with defining $\sin t$ as the unique solution of $x''=-x, x(0)=0,x'(0)=1$.

You get the series by using standard ODE methods

In fact

$x''=-x$ is Sturm-Liouville so you can use all of those things

you get the oscillatory behavior that way.

hmm, no, it's not SL

bolbteppa

The way I remember to derive the integral is to use $\theta = \arcsin(x)$ so that $\sin(\theta) = x$ and you have $\cos(\theta) \frac{d \theta}{d x} = 1$ so that $d \theta = \frac{1}{\cos(\theta)}dx = \frac{1}{\sqrt{1-\sin^2(\theta)}}dx = \frac{1}{\sqrt{1-x^2}}dx$ so that $\theta = \arcsin(x) = \int_0^x \frac{1}{\sqrt{1-t^2}}dt$, you can get it just by drawing the circle and using the arc length formula too I think

0celo7

I think getting periodicity from the power series is hard

bolbteppa

Another way to get this starting definition is to use $s = r \theta$ (can't see a way to avoid this) and the arc length method $ds = \sqrt{dx^2 + dy^2} = \sqrt{1 + (\frac{dy}{dx})^2}dx$ so that for a circle of radius $1$, $s = \theta, ds = d \theta$ and $x^2 + y^2 =1 \rightarrow y = \sqrt{1 - x^2}$ along with $y' = \frac{- x}{\sqrt{1-x^2}}$ implies $ds = \sqrt{1 + \frac{x^2}{(1 - x^2)}} = \sqrt{\frac{1-x^2 + x^2}{1 - x^2}} = \frac{1}{\sqrt{1-x^2}} = d\theta$ so that $\theta = \int d \theta$

There's nothing wrong with defining $\sin$ and $\cos$ that way using differential equations

There is even an amazing visual way to link it to circles using complex differential equations in Tristan's book

0celo7

1:41 AM

how do you get periodicity?

bolbteppa

If you define it using ODE's then the domain of definition is $[0,1]$ and you have no idea what's beyond that, then you simply define the "periodic extension" of $\sin$ to satisfy periodicity by definition

0celo7

why $[0,1]$

bolbteppa

Lets see, you can define most functions a) analytically using integrals, b) iteratively/approximation-eley using series expansions, c) geometrically using graphs/pictures, d) constrainedly using differential equations, e) producteley using infinite products, anything else?

0celo7

c) is bullshit

a) and b) are probably the same?

bolbteppa

Sorry $[-1,1]$

0celo7

1:48 AM

why $1$ tho

bolbteppa

b) is a Taylor/Laurent/Fourier/AHH expansion of a)

Trigonometric functions are also called "circular functions", they are defined with respect to the unit circle $x^2 + y^2 = 1$

The domain of definition of $x$ is $[-1,1]$ via $y = \pm \sqrt{1-x^2}$

0celo7

what

bolbteppa

(Hyperbolic functions are "complex circular functions")

0celo7

I meant using the ODE definition

bolbteppa

Why do you use $[-1,1]$? Because this gives the right answer

If you use $[a,b]$ then you'll get $\sin(\omega x)$ right?

For some $\omega$

(where you define the boundary conditions in the middle of the interval)

0celo7

1:54 AM

dunno

bolbteppa

Alright, well if you visualize the complex number $z = x + iy$ as an arrow, i.e. a vector $z = (x,y)$ in the plane, then you know $z'(t) = \frac{d}{dx}(x(t),y(t))$ is a tangent vector, orthogonal to $z$ right? Then $z''$ is orthogonal to $z'$, i.e. $z''$ is parallel to $z$, that is $z''$ is a multiple of $z$. In general you'll have $z'' = \lambda z$, and you want to find the arrow $z$ such that it's second derivative is a multiple of what it was originally, $z'' = \lambda z$.

What about the special case where $z'' = - z$?

Can you visualize that

What would the motion look like in this case?

What figure would it generate (if there was any motion at all, what is the only way the arrow can move?)?

0celo7

2:34 AM

@dmckee Do you know Sakurai's QM book?

dmckee

@0celo7 No. Keep meaning to look at it. In my copious spare time.

0celo7

@dmckee You're the first prof I've met who has copious spare time

I'm taking a course on QM based out of it

I'm currently trying to decipher physicist mumbo-jumbo regarding the math

also it has really wide margins and sticks out in my bookshelf >:(

what is it with physics books labeling every equation

bolbteppa

2:51 AM

@0celo7 can you prove $\lim_{\theta \rightarrow 0} \frac{\sin(\theta)}{\theta} = 1$?

dmckee

@0celo7 In Jackson's case he's anticipating referring back to them ten chapters later.

bolbteppa

(Obligatory: youtube.com/watch?v=mm-4PltMB2A )

DanielSank

@0celo7 Why?

bolbteppa

@0celo7 page 2 web.auburn.edu/holmerr/1617/Textbook/derivsincos-screen.pdf has the nicest pic of $\sin(\theta) < \theta < \tan(\theta)$ I can find, the proof is immediate from this, that's the hardest part of the proof done

I mean you can even see why the limit is 1 looking at this pic

0celo7

@DanielSank why what?

On mobile

@bolbteppa No, not without using sin(x)<x, which I have no reason to believe.

bolbteppa

3:06 AM

Can you not see the picture in that link?

0celo7

Picture, smicture

bolbteppa

Using $s = r \theta$ with $r = 1$ you have $s = \theta$ and so in the picture you immediately see $\sin(\theta) < \theta < \tan(\theta)$

0celo7

I'm talking analytically

bolbteppa

What does analytically mean?

0celo7

Using the ODE/power series def

No picture

I'm going to sleep, cheerio

bolbteppa

3:09 AM

Getting a bit circular now

You have some random power series which you randomly labelled sin and cos, the derivative of one gives the other instantly, so you're done. Why you called those power series sin and cos, and how those random power series link to circles, has yet to be explained by you.

If you want to use those in the limit proof above, you have to make sure those crazy power series satisfy the trigonometric addition theorem too, then those limits should be obvious immediately, again to be explained by you, don't know what the issue is anymore

I recommend putting down the QM or GR books and picking up the calculus books, no reason why you can't zoom though them if Wald is so easy

Sir Cumference

5:17 AM

You know what I love about the Universe, if it's infinite? There are a finite number of quantum states that matter in a given volume can have.

So if you travelled far enough, you'd start seeing repetitions of our observable universe.

Meaning that there would be observable universes slightly different from ours. We basically have a multiverse inside a universe.

Secret

5:33 AM

That's a multiverse level 1, and what you said is related to poncaire recurrence

peterh

5:45 AM

@ACuriousMind If you found my opinion false, you could correct it, you didn't do. I want to understand you, I explained it not once but many times. You choose to go into silence, and I not "I am not listening".

@ACuriousMind You are over-sensitive, I never had any intent to accuse you, or have any conflict with you! I want(ed) only to understand your vision and... well... voting patterns. If I formulated incorrectly, it is my mistake, and I am sorry for that. But I am sure, the statement that "I am not listening", is simply exactly the opposite of the reality.

peterh

6:03 AM

@knzhou On this ACM thread, please read my previous 2 chat messages. I still honor ACM, the physicist. I simply can't understand why do you consider "hate" that I am just curious to somebodys views. Now I think the best if also I prefer a read-only mode here, although I am a little bit disappointed in the treatment I've got.

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