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

## Kronecker delta / please help me with my homework

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- Cosmologicon
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### Re: Kronecker delta / please help me with my homework

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?

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.

- Cosmologicon
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- 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.

Hint: summation and integration commute.

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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.

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

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