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- 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
- Eigenvalues (
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
- Projective Orthonormality:
- Complex Conjugation:
- 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)
- For
any function of
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
- 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
- This puts the VSHs into vector form:
- Hansen Multipole Properties
_L & = & 0
_L & = & 0
_L & = & i k f_&ell#ell;(kr) Y_L()
_L & = & -ik _L
_L & = & ik _L
_L & = & 0
- 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: Optical Scattering
Up: The Hansen Multipoles
Previous: Concluding Remarks About Multipoles
Contents
Robert G. Brown
2017-07-11