- #1

Yoran91

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

[itex]\mathscr{D} T^{k}_{q} \mathscr{D}^{\dagger} = \sum_{q'} \mathscr{D}^{k}_{q' q} T^{k}_{q'} [/itex]

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

Now I wish to prove that [itex]X^{k_1}_{q_1} = T^{k_1}_{q_1} \otimes \mathbb{1} [/itex] is an irreducible spherical tensor operator whenever [itex]T^{k_1}_{q_1} [/itex] is.

Thus, I wish to prove

[itex]\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'} [/itex].

I know that [itex]\mathscr{D}= \mathscr{D}_1 \otimes \mathscr{D}_2 [/itex] 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

[itex]\mathscr{D} X^{k_1}_{q_1} \mathscr{D}^{\dagger} = \sum_{q_1'} \mathscr{D}^{k_1}_{1 q_1' q_1} [/itex],

with the wigner D matrix D_1 on the right hand side instead of the regular one.

Can anyone help with this?