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{SECT 0 {EXCHG {PARA 258 "" 0 "" {TEXT -1 42 "LINEAR COMBINATIONS OF S
PHERICAL HARMONICS" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 264 35 "by Kerry
 Browne and Corinne Manogue" }}{PARA 256 "" 0 "" {TEXT -1 30 "Copyrigh
t 2006 Corinne Manogue" }{TEXT 265 1 "\n" }}{PARA 256 "" 0 "" {TEXT 
-1 78 "In this worksheet, you will examine combinations of the spheric
al \nharmonics, " }{XPPEDIT 18 0 "Y[l,m](theta,phi);" "6#-&%\"YG6$%\"l
G%\"mG6$%&thetaG%$phiG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 147 "restart:with(plots):with(orthopoly):\nsetoptions3d(a
xes=boxed,scaling=constrained,\nstyle=patchnogrid):\nassume(theta, rea
l, phi, real):\nnumpts:=8000:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 257 20 "First we define the " }{XPPEDIT 
258 0 "Y[l,m](theta,phi);" "6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$phiG" }
{TEXT 256 43 " just as we did in the previous worksheet. " }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "Y:= proc(l,m)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "local p, Ptheta, nrm:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "p:
=(1-x^2)^(abs(m)/2)*diff(P(l,x),[x$abs(m)]):" }{TEXT -1 0 "" }}{PARA 
257 "> " 0 "" {MPLTEXT 1 0 29 "Ptheta:=subs(x=cos(theta),p):" }}{PARA 
257 "> " 0 "" {MPLTEXT 1 0 87 "nrm:=(-1)^((m+abs(m))/2)*sqrt((2*l+1)\n
*factorial(l-abs(m))/(4*Pi*factorial(l+abs(m)))):" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 28 "nrm*Ptheta*exp(I*m*phi)\nend:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 259 17 "\nChoose a sum of " }{XPPEDIT 263 0 "Y[l,m](theta
,phi)" "6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$phiG" }{TEXT 260 2 "'s" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ylmsum:=??;" }}}{EXCHG 
{PARA 256 "" 0 "" {TEXT -1 24 "\nNext, \"square\" the sum " }{TEXT 
261 3 "of " }{XPPEDIT 18 0 "Y[l,m](theta,phi);" "6#-&%\"YG6$%\"lG%\"mG
6$%&thetaG%$phiG" }{TEXT 262 2 "'s" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "ylmsumsq:=(simplify(conjugate(ylmsum)*ylmsum));" }}}{EXCHG {PARA 
256 "" 0 "" {TEXT -1 5 "Plot " }{XPPEDIT 18 0 "abs(Y[l,m](theta,phi))^
2;" "6#*$-%$absG6#-&%\"YG6$%\"lG%\"mG6$%&thetaG%$phiG\"\"#" }{TEXT -1 
78 " on a sphere, coloring the surface with the \nvalue of the probabi
lity density." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "sphereplot(1, phi=
0..2*Pi, theta=0..Pi, \ncolor=ylmsumsq, numpoints=numpts);" }}{PARA 
13 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 
"sphereplot(ylmsumsq,phi=0..2*Pi, theta=0..2*Pi, \ncolor=ylmsumsq,\nnu
mpoints=3*numpts,style=patchnogrid,\nscaling=constrained, axes=boxed);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 0 }
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