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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. :

http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm

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 :)

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# Commutate relation of lowering operator and sperical tensor operators

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