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Commutate relation of lowering operator and sperical tensor operators

  1. Dec 8, 2008 #1
    Hi all,

    I found a commutation relation of lowering operator(J-) and spherical operator in Shankar's QM (2ed, page 418, Eq 15.3.11):
    [tex][J_-,T_k^q] = - \hbar \sqrt{(k+q)(k-q+1)} T_k^{q-1}[/tex]

    I wonder how the minus sign in the begining of the right hand side come out?

    I have googled some pages, some of them have that "-", e.g. :
    the formula appears at the end of this page.

    and some has no "-", e.g.:
    http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/node30.html [Broken]
    Eq(116) at the begining .

    I also found there's no minus in Messiah's QM (Vol II, page 572, Eq XIII.123a)
    [tex][J_-,T_q^{(k)}] = \sqrt{k(k+1)-q(q-1)} T_{q-1}^{(k)}[/tex]

    So, the question is which one is correct?

    Thanks :)
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Dec 9, 2008 #2


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    Science Advisor

    Both are correct. It depends on the convention adopted for the T operators. In particular, for a vector operator, it depends on whether [tex]T^{\pm 1}_1 = x\pm iy[/tex] or [tex]T^{\pm 1}_1 = \pm(x\pm iy)[/tex].
    Last edited: Dec 9, 2008
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