Parity of the Spherical Harmonics

In spherical coordinates, the parity operation is

$$\begin{eqnarray*} r&\rightarrow& r \\ \theta&\rightarrow& \pi-\theta \\ \phi&\rightarrow& \phi+\pi .\\ \end{eqnarray*}$$

The radial part of the wavefunction, therefore, is unchanged and the

$$\bgroup\color{black} R(r)\rightarrow R(r) \egroup$$

parity of the state is determined from the angular part. We know the state $Y_{\ell\ell}$ in general. A parity transformation gives.

$$\bgroup\color{black} Y_{\ell\ell}(\theta,\phi)\rightarrow Y_{... ...\theta)=e^{i\ell\pi}Y_{\ell\ell}=(-1)^\ell Y_{\ell\ell} \egroup$$

The states are either even or odd parity depending on the quantum number $\ell$.

$$\bgroup\color{black} parity=(-1)^\ell \egroup$$

The angular momentum operators are axial vectors and do not change sign under a parity transformation. Therefore, $L_-$ does not change under parity and all the $Y_{\ell m}$ with have the same parity as $Y_{\ell\ell}$

$$\begin{eqnarray*} L_-Y_{\ell\ell}\rightarrow (-1)^\ell L_-Y_{\ell\ell} \\ Y_{\ell m}(\pi-\theta,\phi+\pi)=(-1)^\ell Y_{\ell m}(\theta,\phi)\\ \end{eqnarray*}$$