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I fail to follow one step in the process of calculating ⟨l_{a}∥Y^{(L)}∥l_{b}⟩ .

The spherical harmonics Y_{ma}^{(L)}(r) represent the 2L+1 components of the spherical tensor of rank L. Writing the Eckart-Wigner th. for M = 0 yields:

(1)

Also one can write

(2)

Coupling L and l_{b}to l:

(3)

Thus having

(4)

Now solving the integral:

(5)

So:

(6)

Here is my problem! After solving the integral (5) and replacing it into (4) I don't understand how that changes the Wigner 3j symbols from (3) into (6)

Could anyone please help me with this step? I m guessing it has something to do with does kronecker deltas from solving the integral and they act on the wigner symbols after substitution... but i have no idea how!

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# A Calculation a reduced matrix element using E-Wigner Th.

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