Commutate relation of lowering operator and sperical tensor operators

[Einsling]

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):
$$[J_-,T_k^q] = - \hbar \sqrt{(k+q)(k-q+1)} T_k^{q-1}$$

I wonder how the minus sign in the beginning 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
Eq(116) at the beginning .

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


So, the question is which one is correct?

Thanks :)

 

[Avodyne]

Both are correct. It depends on the convention adopted for the T operators. In particular, for a vector operator, it depends on whether $$T^{\pm 1}_1 = x\pm iy$$ or $$T^{\pm 1}_1 = \pm(x\pm iy)$$.