# Spherical tensor operators' commutation with lowering/raising operator

1. Jun 23, 2013

### cattlecattle

I'm studying Shankar's book (2nd edition), and I came across his equation (15.3.11) about spherical tensor operators:
$[J_\pm, T_k^q]=\pm \hbar\sqrt{(k\mp q)(k\pm q+1)}T_k^{q\pm 1}$

I tried to derive this using his hint from Ex 15.3.2, but the result I got doesn't have the overall $\pm$ sign on the RHS (i.e., it's always $+\hbar$ on the RHS). And I think my result agrees with the result of k=1 where $T_1^\pm=\mp\frac{V_x\pm iV_y}{\sqrt{2}},T_1^0=V_z$ as well.

So I thought this was simply a typo from Shankar's book, but it seems all erratas of this book don't list this as a typo. And the wikipedia article and various other online tutorials all agree with Shankar.

However, Sakurai's book agrees with me (second edition, Eq 3.11.25b)

I'm wondering which version is correct, or could it be that the two versions are adopting different conventions. I'm leaning towards the belief that Shankar's version is a typo and all the online tutorials and the wikipedia article are simply copying the same typo.

Last edited: Jun 23, 2013
2. Jun 23, 2013

### Bill_K

My references (Edmonds and Messiah) agree with you.