Math

[eqn]\text{Let } A=\int_{-\infty}^\infty e^{-x^2}dx[/eqn]

[eqn]A^2=A\cdot A=\int_{-\infty}^\infty e^{-x^2}dx\int_{-\infty}^\infty e^{-y^2}dy = \iint_{\mathbb{R}^2} e^{-(x^2+y^2)}dx\ dy[/eqn]

Rewrite in polar coordinates

[eqn]A^2 = \int_0^{\infty} \int_0^{2\pi} e^{-r^2}r\ dr\ d \theta =2\pi \int_0^{\infty} e^{-r^2}r\ dr = 2\pi ( -\frac{1}{2}e^{-r^2})\rvert_{0}^{\infty}=\pi[/eqn]

[eqn]A=\sqrt{\pi}[/eqn]

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OMG that babby is soooo smart!! xDDD

i bet that mom will be so surprised When zhe sees that LOL!!!!!!1

I told you no drawing on the wall!
>Puts baby in crib and washes the wall down.

Is there a scientific reason why moms are always ugly? Why do men keep fucking these hags?

baby want fuck

can u explain the magic behind rewriting shit in polar coordinates? I don't mean the technique of actually doing this (like in your example) as that's quite easy but rather why exactly can i do that?
I'm not a math person but i had this concecpt mentioned in several classes (theoretical chemistry for example) and it was never explained why exactly we can do this, just
"rewrite in polar coordinates".. and i'm quite curious how this works but way to uneducated in math to know where to look for answer to such questions..

What's the 'actual' math behind rewriting ?

Coordinates don't matter. You can choose whatever you want and the answer should be the same.

Rewriting mathematical expressions in an equivalent form often makes them easier to evaluate.

I'm too lazy to come up with an example right now, though.

but how does that let me answer questions i wasn't able to answer beforehand if it doesn't matter which i choose?

why i'm asking is the following:

We pretty much have our usual "problem" -> transform coordinates (which i don't really get the concept of -> solving of the polar coordinates with legendre polynomials (which i also have absolutely no fucking clue about as it's not explained (pic related only explanation there is in my text)

so i don't really get how all these crappy functions relate to my actual problem.. it's just i throw my problem in a machine that works magic and after some 2-3 site long annoying math you actually get a result but have no clue how you got there

Well, polar coordinates aren't magic. You're just rewriting such-and-such expression in a different but equivalent form so it's easier to work with. You might be having other issues if you are just plugging and chugging without understanding what you're doing, though. You definitely ought to review this stuff and make sure you understand it instead of just following a step-by-step procedure. Seek different resources if your textbook is shit. You have the internet at your disposal, after all.

The point of coordinate transformations is to represent the problem in a way that makes it more manageable. Physics (and when denoted in mathematics) is the same in all coordinate systems, it's a very common symmetry that exists. You are not forced to change coordinates but doing so will make a solution more obvious sometimes. The spherical harmonics in your photo come from solving a PDE with boundary conditions set on a spherical surface. I suggest googling it. The l's I'm going to assume are the orbital angular momentum quantum numbers, which as you can guess by the name are related to the spherical harmonics because of the geometry of the problem.

I pretty much understand all the single parts (like 1-4) but i dont see the 'big' picture (5)

>I suggest googling it

Problem here is my textbook and all related resources to passing this goddamn exam are in german and it's next to impossible to search in a foreign language about concepts you barely know in german (i.e. most of the time german google results suck and i have no clue where to look for in english besides taking those 'standard' textbooks that get promoted here all the time and work them through completely

also why exactly am i looking at L2
why not L3 or whatever? Here in my book it's pretty much just
here is L
here is L2
this how to solve L2 or whatever the fuck you call what i am doing at this point
god i hate quantum...

Give me a sec, I can screen shot some derivations from Griffiths if you want.

any help is appreciated :-)

here have a picture of a funny molecule for your efforts

That's one hot delocalized vagoo

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Post time limit sucks.

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>the equation is not so simple

i think i should look into this 'seperation of variables' as this is the part where i get lost in my book and also get lost in your book.
Never heard the concept of a seperation constant which is pretty much needed i guess

but with those pics, i know where to start - thank you Veeky Forums bro

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Last page. More reference: math.cornell.edu/~rand/randdocs/PDE_handout/PDE15.pdf

And no problem man.

>he's not a milfhunter

There's a difference between a mathematical statement and a mathematical problem dude.

Very nice.

It's a standard worked out problem in a lot of textbooks but he had nice latex work I guess

The latex was shitty, he forgot to size his brackets properly.

Well I guess it was a gay meme-post with no redeeming qualities then.

L2 is an inner-product space

Is there a physical interpretation for L2? I can see it has some nice properties and is "easy" to calculate.
I can pretty much imagine what angular momentum is but i don't know how think about L2..
Also is this somehow related to the probability amplitude (where i also don't know why i actually use the squared function) ?

I have a master's in bioethics and what is this?

The physical interpretation of L^2 is straightforward, it's the squared magnitude of the angular momentum.

The nice thing about it is that it commutes with all the L_i's, allowing you to construct a nice basis for your Hilbert space. In fact, you can calculate all the eigenvalues of L^2 and L_z (the L_i's are incompatible observables, i.e. they don't commute, and therefore you can't diagonalize all of them simultaneously), and therefore know the possible outcomes of their measurement, by simply studying the commutation relations of the L^2, L_z and the raising and lowering operators and their action on your Hilbert space, there is no need of solving any differential equations (as long as you don't ask for the wave functions). In more technical terms, you just have to study the representations of the algebra defined by your L_i's on your Hilbert space, which then turns out to be a highest weight representation involving the L_z and the lowering & raising operators (from which you can then find L_x and L_y).

It's related to the probability amplitude because the eigenfunctions (which you're finding by solving the differential equations) are by definition the probability amplitudes that you're going to obtain a certain value for the L^2 and L_z after measurement, written in position space. The square of the probability amplitude is, by definition, the probability density. These final points are included in the postulates of QM, stop being such a shithead.

They become uglier after being moms so other men leave them alone