Kronecker delta / please help me with my homework

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Kronecker delta / please help me with my homework

Postby Mat » Wed May 09, 2007 12:59 pm UTC

Hi, this is probably a stupid question, but I suck at maths :(

in a problem sheet for atomic physics I've been told to use:
the integral of Y*_lm Y_l'm' sin(theta) dthetadphi = kronecker delta_l,l' kronecker delta m,m'

I get that kronecker delta of l,m = 1 if theyre equal, 0 otherwise, but I don't understand what the subscripts mean in this case, how are l,m related to l prime and m prime?

In my question I have a sum over a (from 0 to infinity), and a sum over b (from -a to a) of Y*_ab(theta1,phi1) Y_ab(theta2,phi2) which I'm then integrating over two volumes, dV_1,dV_2. (theres some other stuff as well, that depends on r1 and r2) How do I rewrite this Y*Y part in terms of delta functions? It would be nice if that whole part was just delta_ab but I can't just write that down. :(

*edit* this is what I have to work out:
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Postby Gelsamel » Wed May 09, 2007 1:23 pm UTC

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Re: Kronecker delta / please help me with my homework

Postby Cosmologicon » Wed May 09, 2007 2:33 pm UTC

Mat wrote:How do I rewrite this Y*Y part in terms of delta functions? It would be nice if that whole part was just delta_ab but I can't just write that down. :(

I don't remember my Bessel functions as well as I should, but it's probably not that. You should rewrite Y* in terms of Y. Look at the form of Y_ab and see what happens if you complex-conjugate it. You may even have an identity that shows you. I'm just throwing out an example here, but it might be something like:

Y*_ab = Y_(-a)b if b is even
Y*_ab = (-1)^a Y_ab if b is odd

You know?
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Postby Mat » Wed May 09, 2007 3:18 pm UTC

I don't think I know what Y_ab actually is. :?

I think I may have done it by writing out the theta_1,phi_1 bits seperately from the theta_2 phi_2 bits and writing the psi_100 as a function of Y_00 so I got Y*_ab Y_00 sin theta1 and Y*_00 Y_ab sin theta2, which are the right form for the equation they told me to use, at least, and make the horrible sums ago away.
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Postby Mat » Wed May 09, 2007 3:18 pm UTC

Gelsamel wrote:The Cat in your avvy is so cute. It looks like my cat :-).

:)
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Postby Cosmologicon » Wed May 09, 2007 3:22 pm UTC

Well it can't be that hard to find out what Y_ab is. I know I've seen the function before, but I can't remember the name. If you give the name it should be easy to search for.
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Postby Mat » Wed May 09, 2007 3:53 pm UTC

I don't think it represents a fancy function I've never heard off, I thought it was part of a wave function? I think the subscript letters represent quantum numbers, but I'm not entirely sure - I don't really understand the physical significance of it all.
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Postby Cosmologicon » Wed May 09, 2007 4:29 pm UTC

Okay, sorry, I was completely misreading. You're right, you don't need the form of Y_ab. You're also right that you can't just replace Y*Y with delta functions. You need to have the integral around it too. So replace the 1/r_12 in your integral there with the summation you're given, and see if you can manipulate it so that you have the integral over Y*Y.

Hint: summation and integration commute.
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Postby adlaiff6 » Wed May 09, 2007 4:54 pm UTC

Looks like my old cat too, before he got fat. Now he's slimming down again though.
3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n
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Postby skeptical scientist » Wed May 09, 2007 10:00 pm UTC

Cosmologicon wrote:Okay, sorry, I was completely misreading. You're right, you don't need the form of Y_ab. You're also right that you can't just replace Y*Y with delta functions. You need to have the integral around it too. So replace the 1/r_12 in your integral there with the summation you're given, and see if you can manipulate it so that you have the integral over Y*Y.

Hint: summation and integration commute.

At least if you're a physicist. If you're a mathematician you actually need to check that the things you are summing and integrating are nice enough first.
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Postby Gelsamel » Thu May 10, 2007 1:43 am UTC

adlaiff6 wrote:Looks like my old cat too, before he got fat. Now he's slimming down again though.


My cat is just huge, he is like a panther. Rawr ~
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Postby Redoubts » Tue May 15, 2007 1:01 am UTC

Kronecker delta l,l', m,m' means that the integral is not zero only if both m = m' AND l = l'

Y is the Legendre polynomial times e^im(phi). However, the form is probably irrelevant to you.

the trick is to note that you can throw away some (most) terms of the series because of the kronecker delta.
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