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

  1. Aug 5, 2017 #1
    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:
    Screen Shot 2017-08-05 at 2.12.25 PM.png (1)

    Also one can write
    Screen Shot 2017-08-05 at 2.13.17 PM.png (2)

    Coupling L and lb to l:
    Screen Shot 2017-08-05 at 2.14.00 PM.png (3)

    Thus having
    Screen Shot 2017-08-05 at 2.14.30 PM.png (4)

    Now solving the integral:
    Screen Shot 2017-08-05 at 2.15.05 PM.png (5)

    So:

    Screen Shot 2017-08-05 at 2.15.31 PM.png (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 Screen Shot 2017-08-05 at 2.17.26 PM.png (3) into Screen Shot 2017-08-05 at 2.17.52 PM.png (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. jcsd
  3. Aug 5, 2017 #2

    blue_leaf77

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    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##.
     
  4. Aug 5, 2017 #3
    Oww i get it! Thank you so much!
     
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