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Table of Properties of Vector Harmonics

  1. Basic Definitions &ell#ell;&ell#ell;m & = &1&ell#ell;(&ell#ell;+1) Y_&ell#ell;,m
    &ell#ell;&ell#ell;-1m & = &-1&ell#ell;(2&ell#ell;+1) [ -&ell#ell; + i×]Y_&ell#ell;,m
    &ell#ell;&ell#ell;+1m & = & -1(&ell#ell;+1)(2&ell#ell;+1) [ (&ell#ell;+1) + i×]Y_&ell#ell;,m

  2. Eigenvalues ($ j,\ell,m$ are integral): J^2 j &ell#ell;m & = & j(j+1)j &ell#ell;m
    L^2 j &ell#ell;m & = & &ell#ell;(&ell#ell;+1)j &ell#ell;m
    J_z j &ell#ell;m & = & m j &ell#ell;m

  3. Projective Orthonormality:

    $\displaystyle \int \vsh{j \ell}{m}\cdot\vsh{j' \ell'}{m' \ast}\ d\Omega =
\delta_{jj'}\delta_{\ell\ell'}\delta_{mm'} \nonumber
$

  4. Complex Conjugation:

    $\displaystyle \vsh{j \ell}{m \ast} = (-1)^{\ell + 1 - j} (-1)^m \vsh{j \ell}{-m}
$

  5. Addition Theorem (LCB notes corrupt - this needs to be checked): j &ell#ell;m &ast#ast;·j' &ell#ell;'m' & = & &sum#sum;_n (-1)^m+1(2&ell#ell;+1)(2&ell#ell;'+1)(2j'+1)(2j+1)4&pi#pi;(2n+1) ×
    & &         C_000^&ell#ell;&ell#ell;'nC_0,-m,m'^jj'n W(j&ell#ell;j'&ell#ell;';n) Y_n,(m'-m)

  6. For $ F$ any function of $ r$ only: (&ell#ell;&ell#ell;mF) & = & 0
    (&ell#ell;&ell#ell;-1mF) & = & &ell#ell;2&ell#ell;+1 [ (&ell#ell;-1)Fr - dFdr] Y_&ell#ell;,m
    (&ell#ell;&ell#ell;+1mF) & = & &ell#ell;+12&ell#ell;+1 [ (&ell#ell;+2)Fr - dFdr] Y_&ell#ell;,m

  7. Ditto: i(&ell#ell;&ell#ell;mF) & = & &ell#ell;+12&ell#ell;+1 [ (&ell#ell;+1)Fr + dFdr]&ell#ell;&ell#ell;-1m + &ell#ell;2&ell#ell;+1 [-&ell#ell;Fr + dFdr]&ell#ell;&ell#ell;+1m
    i(&ell#ell;&ell#ell;-1mF) & = & -&ell#ell;+12&ell#ell;+1 [ (&ell#ell;-1)Fr - dFdr]&ell#ell;&ell#ell;m
    i(&ell#ell;&ell#ell;+1mF) & = & &ell#ell;2&ell#ell;+1 [ (&ell#ell;+2)Fr - dFdr]&ell#ell;&ell#ell;m

  8. This puts the VSHs into vector form:

    \begin{displaymath}
\vsh{\ell \ell}{m} = \left(
\begin{array}{c}
-\sqrt{\frac{(...
...\ell+m+1)}{2\ell(\ell+1)}} Y_{\ell,m+1} \\
\end{array}\right)
\end{displaymath}

    \begin{displaymath}
\vsh{\ell \ell-1}{m} = \left(
\begin{array}{c}
\sqrt{\frac{...
...ell-m)}{2\ell(2\ell-1)}} Y_{\ell-1,m+1} \\
\end{array}\right)
\end{displaymath}

    \begin{displaymath}
\vsh{\ell \ell+1}{m} = \left(
\begin{array}{c}
\sqrt{\frac{...
...1)}{2(\ell+1)(2\ell+3)}} Y_{\ell+1,m+1} \\
\end{array}\right)
\end{displaymath}

  9. Hansen Multipole Properties _L & = & 0
    _L & = & 0
    _L & = & i k f_&ell#ell;(kr) Y_L() _L & = & -ik _L
    _L & = & ik _L
    _L & = & 0

  10. Hansen Multipole Explicit Forms _L & = & f_&ell#ell;(kr) &ell#ell;&ell#ell;m
    _L & = & &ell#ell;+12 &ell#ell;+1 f_&ell#ell;- 1(kr) &ell#ell;, &ell#ell;-1m - &ell#ell;2 &ell#ell;+ 1 f_&ell#ell;+ 1(kr) &ell#ell;,&ell#ell;+1m
    _L & = & &ell#ell;2 &ell#ell;+ 1 f_&ell#ell;- 1(kr) &ell#ell;, &ell#ell;-1m + &ell#ell;+12 &ell#ell;+1 f_&ell#ell;+ 1(kr) &ell#ell;,&ell#ell;+1m _L & = & f_&ell#ell;(kr) &ell#ell;&ell#ell;m
    _L & = & 1kr { d   d(kr) (kr f_&ell#ell;) (i×&ell#ell;&ell#ell;m ) - &ell#ell;(&ell#ell;+ 1) f_&ell#ell;Y_L }
    _L & = & &ell#ell;(&ell#ell;+ 1) 1kr (i ×f_&ell#ell;&ell#ell;&ell#ell;m) - [d   d(kr) f_&ell#ell; ] Y_L


next up previous contents
Next: Optical Scattering Up: The Hansen Multipoles Previous: Concluding Remarks About Multipoles   Contents
Robert G. Brown 2017-07-11