Our angular momentum coupled two-body wave function obeys clearly this definition, namely $$ |(ab)JM\rangle = \left\{a^{\dagger}_aa^{\dagger}_b\right\}^J_M|\Phi_0\rangle=N_{ab}\sum_{m_am_b}\langle j_am_aj_bm_b|JM\rangle|\Phi^{ab}\rangle, $$ is a tensor of rank \( J \) with \( M \) components. Another well-known example is given by the spherical harmonics (see examples during today's lecture).

The product of two irreducible tensor operators $$ T^{\lambda_3}_{\mu_3}=\sum_{\mu_1\mu_2}\langle \lambda_1\mu_1\lambda_2\mu_2|\lambda_3\mu_3\rangle T^{\lambda_1}_{\mu_1}T^{\lambda_2}_{\mu_2} $$ is also a tensor operator of rank \( \lambda_3 \).