I would like to know how to argue if the inner product of two spherical harmonics is zero using symmetry arguments. If the inner product is given by the following integral,
$$\left\langle Y_{\ell}^{m},Y_{j}^{k}\right\rangle=\int_0^{2\pi}\int_0^{\pi} \left(Y_{\ell}^{m}(\theta, \varphi)\right)^*Y_{j}^{k}(\theta, \varphi)\sin\theta\,\mathrm d\theta\,\mathrm d \phi$$
and keeping in mind that theire parity is such that $$
Y_{\ell}^{m}(\pi-\theta, \pi+\phi)=(-1)^{\ell} Y_{\ell}^{m}(\theta, \phi)
$$
How could this feature be used?
Best Answer
It can be shown that the inner product of two spherical harmonics $\left\langle Y_{\ell_1}^{m_1},Y_{\ell_2}^{m_2}\right\rangle$ cancels out whether
The first condition, $m_2 -m_1\in \mathbb{Z}\setminus \{0\}$, arises from the fact that $Y_{\ell}^{m}=\Phi_m(\varphi)\Theta_{l,m}(\theta)$, with $\Phi_m(\varphi)=\frac{1}{\sqrt{2\pi}}e^{im\varphi}$, so
$$\int_0^{2\pi} \left( \Phi_{m_1}(\varphi) \right)^* \Phi_{m_2}(\varphi) \mathrm d \phi= \frac{1}{{2\pi}} \int_0^{2\pi} e^{i(m_2-m_1)\varphi} \mathrm d \phi$$
which is zero if $m_2 -m_1$ is a non-zero integer.
You can get to the second one by making a change of variable $\tilde\theta = \pi-\theta$ and $\tilde\varphi = \pi+\varphi$ in the integral and applying the parity property of the spherical harmonics.