Add a note on various definitions of reduced matrix elements.

chap3/Emissionrates.tex

@@ -135,7 +135,7 @@ \subsection{Polarizability of alkali atoms}
 \hat{\mathbf{d}}_{f'f} &=  \sum_q \sum_{m, m'}  \bra{f' m'} d_{f'\!,f}^q \ket{f m} \op{f' m'}{f m} \mathbf{e}_q^* \\
 & = \bra{ f' }| d | \ket{ f } \sum_q \sum_{m, m'}  \bravket{f m; 1 q}{f' m'} \op{f' m'}{f m} \mathbf{e}_q^* \label{eq:opdff'},
 \end{align}
-where we used the Wigner-Eckart theorem\index{Wigner-Eckart theorem} to pull out the reduced matrix element $\bra{ f' }| d | \ket{ f } $. We denote $ C^{fm;1q}_{f'm'}=  \bravket{f' m'}{f m; 1 q}= \bravket{f m; 1 q}{f' m'}$ as the Clebsch-Gordan coefficients\index{Clebsch-Gordan coefficient}. It can be further simplified with another application of the Wigner-Eckart theorem~\cite{Deutsch2010a},
+where we used the Wigner-Eckart theorem\index{Wigner-Eckart theorem} to pull out the reduced matrix element $\bra{ f' }| d | \ket{ f } $\footnote{In literature, the Wigner-Eckart theorem is expressed in different ways, which leads to different forms of the reduced matrix elements. For instance, our definition of the reduced matrix elements in fine structure levels has a factor of $ \sqrt{2j'+1} $ difference from Ref.~\cite{Lacroute2012} (after correction) and Ref.~\cite{LeKien2013}.}. We denote $ C^{fm;1q}_{f'm'}=  \bravket{f' m'}{f m; 1 q}= \bravket{f m; 1 q}{f' m'}$ as the Clebsch-Gordan coefficients\index{Clebsch-Gordan coefficient}. It can be further simplified with another application of the Wigner-Eckart theorem~\cite{Deutsch2010a},
 	\begin{align}
 		\bra{ f' }| d | \ket{ f } = \bra{ j' }| d | \ket{ j } o^{j' f'}_{j f},
 	\end{align}