Parity of the Spherical Harmonics

In spherical coordinates, the parity operation is

r&\rightarrow& r \\
\theta&\rightarrow& \pi-\theta \\
\phi&\rightarrow& \phi+\pi .\\

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

\begin{displaymath}\bgroup\color{black} R(r)\rightarrow R(r) \egroup\end{displaymath}

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

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

The states are either even or odd parity depending on the quantum number \bgroup\color{black}$\ell$\egroup.

\begin{displaymath}\bgroup\color{black} parity=(-1)^\ell \egroup\end{displaymath}

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

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)\\

Jim Branson 2013-04-22