Stuck on proof irreducible spherical tensor operator

[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?

 

[eljose79]

Hi there,

I understand your frustration with proving this statement. It can be a bit tricky to apply the Wigner D matrices in this context. However, I believe I can offer some help.

Firstly, it's important to note that the Wigner D matrix for a tensor product space is actually a tensor product of the individual Wigner D matrices. In other words, \mathscr{D}= \mathscr{D}_1 \otimes \mathscr{D}_2 means that the Wigner D matrix for the tensor product space is the tensor product of the Wigner D matrices for the individual spaces. This is important to keep in mind when applying the Wigner D matrices to the formula you wish to prove.

Now, let's take a closer look at the formula you wish to prove. You have correctly applied the Wigner D matrix to the left hand side, but on the right hand side, you need to take into account the tensor product structure of X^{k_1}_{q_1}. This means that you need to apply the Wigner D matrix to both T^{k_1}_{q_1} and \mathbb{1}. This will result in the Wigner D matrix for the tensor product space, which you can then use to rewrite the formula as follows:

\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\dagger} = \sum_{q_1'} \mathscr{D}^{k_1}_{q_1' q_1} (\mathscr{D}_1 \otimes \mathscr{D}_2) T^{k_1}_{q_1'}.

Now, using the definition of the Wigner D matrix for a tensor product space, we can rewrite this as:

\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\dagger} = \sum_{q_1'} \mathscr{D}^{k_1}_{q_1' q_1} (\mathscr{D}_1 T^{k_1}_{q_1'} \otimes \mathscr{D}_2).

Finally, using the definition of X^{k_1}_{q_1}, we can rewrite this as:

\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\d