# Stuck on proof irreducible spherical tensor operator

1. May 22, 2013

### Yoran91

Hi everyone,

I'm stuck on proving that a certain operator is an irreducible spherical tensor operator. These are tensor operators $T^{k}_{q}$ with $-k \leq q\leq k$ satisfying

$\mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'}$

where $\mathscr{D}\left(\hat{n},\phi\right) =\exp\left[-\frac{i\phi}{\hbar} \vec{J}\cdot \hat{n} \right]$ is the Wigner-D matrix and $\mathscr{D}^k_{q' q}$ are its matrix elements with respect to the angular momentum basis $| k q \rangle$.

Now I wish to prove that $X^{k_1}_{q_1} = T^{k_1}_{q_1} \otimes \mathbb{1}$ is an irreducible spherical tensor operator whenever $T^{k_1}_{q_1}$ is.
Thus, I wish to prove
$\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\dagger} = \sum_{q_1'} \mathscr{D}^{k_1}_{q_1' q_1} X^{k_1}_{q_1'}$.

I know that $\mathscr{D}= \mathscr{D}_1 \otimes \mathscr{D}_2$ in terms of the Wigner D matrices on the components of the tensor product HIlbert space, but I can't seem to apply this to get the above formula. All I can get is
$\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\dagger} = \sum_{q_1'} \mathscr{D}^{k_1}_{1 q_1' q_1}$,

with the wigner D matrix D_1 on the right hand side instead of the regular one.
Can anyone help with this?