I am recently reading about the Wigner-Eckart and Clebsch-Gordon sections in Sakurai, Modern Quantum Mechanics, 2nd Ed (2014).
On p. 234, Eq. (8.72), I found that he derived the expression for the product of two spherical harmonics:
$$Y^{m_1}_{l_1} Y^{m_2}_{l_2} = \frac{\sqrt{(2l_1+1)(2l_2+1)}}{4\pi}\sum_{l'}\sum_{m'} \langle l_1, l_2, m_1, m_2 | l', m' \rangle \langle l_1, l_2, 0, 0 | l', 0 \rangle \sqrt{\frac{4\pi}{2l'+1}} Y^{m'}_{l'}, $$
where the bracket factors are the usual Clebsch-Gordon coefficients.
Though it might be really obvious, I still want to ask whether this simply means that the product of two spherical harmonics $Y^{m_1}_{l_1} Y^{m_2}_{l_2}$ is not equal to the direct product of the corresponding angular momentum basis vectors $|l_1, m_1\rangle \otimes |l_2, m_2\rangle$? And if it is the case, then may I ask why or how should I comprehend this? As I just so naively thought they should be the same thing for quite a while given $|l, m\rangle = Y^m_l$.
