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Table of Properties of Vector Harmonics

  1. Basic Definitions
    $\displaystyle \mbox{\boldmath$Y$}_{\ell \ell}^{m}$ $\textstyle =$ $\displaystyle \frac{1}{\sqrt{\ell(\ell+1)}} \mbox{\boldmath$L$}Y_{\ell,m}$  
    $\displaystyle \mbox{\boldmath$Y$}_{\ell \ell-1}^{m}$ $\textstyle =$ $\displaystyle -\frac{1}{\sqrt{\ell(2\ell+1)}} \left[ -\ell
\hat{\mbox{\boldmath$r$}}+ i\hat{\mbox{\boldmath$r$}}\times\mbox{\boldmath$L$}\right]Y_{\ell,m}$  
    $\displaystyle \mbox{\boldmath$Y$}_{\ell \ell+1}^{m}$ $\textstyle =$ $\displaystyle -\frac{1}{\sqrt{(\ell+1)(2\ell+1)}} \left[
(\ell+1) \hat{\mbox{\boldmath$r$}}+ i\hat{\mbox{\boldmath$r$}}\times\mbox{\boldmath$L$}\right]Y_{\ell,m}$  

  2. Eigenvalues ($j,\ell,m$ are integral):
    $\displaystyle J^2 \mbox{\boldmath$Y$}_{j \ell}^{m}$ $\textstyle =$ $\displaystyle j(j+1)\mbox{\boldmath$Y$}_{j \ell}^{m}$  
    $\displaystyle L^2 \mbox{\boldmath$Y$}_{j \ell}^{m}$ $\textstyle =$ $\displaystyle \ell(\ell+1)\mbox{\boldmath$Y$}_{j \ell}^{m}$  
    $\displaystyle J_z \mbox{\boldmath$Y$}_{j \ell}^{m}$ $\textstyle =$ $\displaystyle m \mbox{\boldmath$Y$}_{j \ell}^{m}$  

  3. Projective Orthonormality:

    \begin{displaymath}
\int \mbox{\boldmath$Y$}_{j \ell}^{m}\cdot\mbox{\boldmath$Y$...
...d\Omega =
\delta_{jj'}\delta_{\ell\ell'}\delta_{mm'} \nonumber
\end{displaymath}

  4. Complex Conjugation:

    \begin{displaymath}
\mbox{\boldmath$Y$}_{j \ell}^{m \ast} = (-1)^{\ell + 1 - j} (-1)^m \mbox{\boldmath$Y$}_{j \ell}^{-m}
\end{displaymath}

  5. Addition Theorem (LCB notes corrupt - this needs to be checked):
    $\displaystyle \mbox{\boldmath$Y$}_{j \ell}^{m \ast}\cdot\mbox{\boldmath$Y$}_{j' \ell'}^{m'}$ $\textstyle =$ $\displaystyle \sum_n
(-1)^{m+1}\sqrt{\frac{(2\ell+1)(2\ell'+1)(2j'+1)(2j+1)}{4\pi (2n+1)}}
\times$  
        $\displaystyle \quad\quad C_{000}^{\ell\ell'n}C_{0,-m,m'}^{jj'n}
W(j\ell j'\ell';n) Y_{n,(m'-m)}$  

  6. For $F$ any function of $r$ only:
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot (\mbox{\boldmath$Y$}_{\ell \ell}^{m}F)$ $\textstyle =$ $\displaystyle 0$  
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot (\mbox{\boldmath$Y$}_{\ell \ell-1}^{m}F)$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell}{2\ell+1}} \left[
(\ell-1)\frac{F}{r} - \frac{dF}{dr}\right] Y_{\ell,m}$  
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot (\mbox{\boldmath$Y$}_{\ell \ell+1}^{m}F)$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell+1}{2\ell+1}} \left[
(\ell+2)\frac{F}{r} - \frac{dF}{dr}\right] Y_{\ell,m}$  

  7. Ditto:
    $\displaystyle i\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$Y$}_{\ell \ell}^{m}F)$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell+1}{2\ell+1}} \left[
(\ell+1)\frac{F}{r} + \frac{...
...ft[-\ell\frac{F}{r} +
\frac{dF}{dr}\right]\mbox{\boldmath$Y$}_{\ell \ell+1}^{m}$  
    $\displaystyle i\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$Y$}_{\ell \ell-1}^{m}F)$ $\textstyle =$ $\displaystyle -\sqrt{\frac{\ell+1}{2\ell+1}} \left[
(\ell-1)\frac{F}{r} - \frac{dF}{dr}\right]\mbox{\boldmath$Y$}_{\ell \ell}^{m}$  
    $\displaystyle i\mbox{\boldmath$\nabla$}\times (\mbox{\boldmath$Y$}_{\ell \ell+1}^{m}F)$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell}{2\ell+1}} \left[
(\ell+2)\frac{F}{r} - \frac{dF}{dr}\right]\mbox{\boldmath$Y$}_{\ell \ell}^{m}$  

  8. This puts the VSHs into vector form:

    \begin{displaymath}
\mbox{\boldmath$Y$}_{\ell \ell}^{m} = \left(
\begin{array}{c...
...\ell+m+1)}{2\ell(\ell+1)}} Y_{\ell,m+1} \\
\end{array}\right)
\end{displaymath}


    \begin{displaymath}
\mbox{\boldmath$Y$}_{\ell \ell-1}^{m} = \left(
\begin{array}...
...ell-m)}{2\ell(2\ell-1)}} Y_{\ell-1,m+1} \\
\end{array}\right)
\end{displaymath}


    \begin{displaymath}
\mbox{\boldmath$Y$}_{\ell \ell+1}^{m} = \left(
\begin{array}...
...1)}{2(\ell+1)(2\ell+3)}} Y_{\ell+1,m+1} \\
\end{array}\right)
\end{displaymath}

  9. Hansen Multipole Properties
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$M$}_L}$ $\textstyle =$ $\displaystyle 0$  
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$N$}_L}$ $\textstyle =$ $\displaystyle 0$  
    $\displaystyle \mbox{\boldmath$\nabla$}\cdot {\mbox{\boldmath$L$}_L}$ $\textstyle =$ $\displaystyle i k f_\ell(kr) Y_L(\hat{\mbox{\boldmath$r$}})$  


    $\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$M$}_L}$ $\textstyle =$ $\displaystyle -ik \mbox{\boldmath$N$}_L$  
    $\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$N$}_L}$ $\textstyle =$ $\displaystyle ik \mbox{\boldmath$M$}_L$  
    $\displaystyle \mbox{\boldmath$\nabla$}\times {\mbox{\boldmath$L$}_L}$ $\textstyle =$ $\displaystyle 0$  

  10. Hansen Multipole Explicit Forms
    $\displaystyle \mbox{\boldmath$M$}_L$ $\textstyle =$ $\displaystyle f_\ell(kr) \mbox{\boldmath$Y$}_{\ell \ell}^{m}$  
    $\displaystyle \mbox{\boldmath$N$}_L$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell+1}{2 \ell +1}} f_{\ell - 1}(kr)
\mbox{\boldmath...
...frac{\ell}{2 \ell + 1}} f_{\ell + 1}(kr)
\mbox{\boldmath$Y$}_{\ell,\ell+1}^{m}$  
    $\displaystyle \mbox{\boldmath$L$}_L$ $\textstyle =$ $\displaystyle \sqrt{\frac{\ell}{2 \ell + 1}} f_{\ell - 1}(kr)
\mbox{\boldmath$...
...rac{\ell+1}{2 \ell +1}} f_{\ell + 1}(kr)
\mbox{\boldmath$Y$}_{\ell,\ell+1}^{m}$  


    $\displaystyle \mbox{\boldmath$M$}_L$ $\textstyle =$ $\displaystyle f_\ell(kr) \mbox{\boldmath$Y$}_{\ell \ell}^{m}$  
    $\displaystyle \mbox{\boldmath$N$}_L$ $\textstyle =$ $\displaystyle \frac{1}{kr} \bigg\{ \frac{d~~~}{d(kr)} (kr f_\ell)
\big(i\hat{...
...{m} \big) - \hat{\mbox{\boldmath$r$}}\sqrt{\ell(\ell + 1)}
f_\ell Y_L \bigg\}$  
    $\displaystyle \mbox{\boldmath$L$}_L$ $\textstyle =$ $\displaystyle \sqrt{\ell(\ell + 1)} \frac{1}{kr} (i \hat{\mbox{\boldmath$r$}}
\...
...ll}^{m}) - \hat{\mbox{\boldmath$r$}}\bigg[\frac{d~~~}{d(kr)} f_\ell
\bigg] Y_L$  


next up previous contents
Next: Optical Scattering Up: The Hansen Multipoles Previous: Concluding Remarks About Multipoles   Contents
Robert G. Brown 2014-08-19