For the discussion of math. Duh.

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Mat
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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:

where

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Cosmologicon
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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?

Mat
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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.

Mat
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Cosmologicon
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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.

Mat
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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.

Cosmologicon
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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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3.14159265... wrote:What about quantization? we DO live in a integer world?

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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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Redoubts
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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.