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

1. Aug 5, 2017

squareroot

Hello.

I fail to follow one step in the process of calculating ⟨la∥Y(L)∥lb⟩ .

The spherical harmonics Yma(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 lb 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!

2. Aug 5, 2017

blue_leaf77

In equation (4), the right hand side is summed over $l$ and $m$. However upon inserting the result from equation (5), only the term with $l=l_a$ and $m=-m_a$ survives. Yes it has to do with the property of Dirac delta symbol which is for any pair of integer $i$ and $j$, the Dirac delta $\delta_{ij}$ will give zero if $i\neq j$. If $i=j$, $\delta_{ij} = 1$.

3. Aug 5, 2017

squareroot

Oww i get it! Thank you so much!