{VERSION 6 0 "IBM INTEL NT" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 1 16 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 0 1 0 0 0 
0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
290 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 291 "" 0 1 0 0 
0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim
es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }
{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 
2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 27 "Clebsch-Gordan coefficien
ts" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "The
 calculation of vector coupling coefficients is based on representing \+
J_total -sqared = " }{TEXT 291 1 "J" }{TEXT -1 185 "^2 in the basis of
 the single-particle angular momentum eigenstates and diagonalization.
 We make use of the matrix elements of the raising and lowering operat
ors to assemble the matrix." }}{PARA 0 "" 0 "" {TEXT 19 62 "<j,m+1| J+
 |j,m> = <j,m| J- | j,m+1> = -h_*sqrt((j-m)*(j+m+1))" }}{PARA 0 "" 0 "
" {TEXT -1 31 "We are using the decomposition " }{TEXT 263 1 "J" }
{TEXT -1 9 "_tot^2 = " }{TEXT 264 2 "J1" }{TEXT -1 7 "^2  +  " }{TEXT 
265 2 "J2" }{TEXT -1 9 "^2  +  2 " }{TEXT 266 2 "J1" }{TEXT -1 5 "_z *
 " }{TEXT 267 2 "J2" }{TEXT -1 5 "_z + " }{TEXT 270 2 "J1" }{TEXT -1 
4 "_ * " }{TEXT 269 3 "J2+" }{TEXT -1 5 "  +  " }{TEXT 268 2 "J1" }
{TEXT -1 1 "\000" }{TEXT 271 2 "+ " }{TEXT -1 3 "*  " }{TEXT 272 3 "J2
_" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "Whe
n assembling the matrix representation we keep in mind that the 5 term
s on the RHS of the above equation separate into two groups:" }}{PARA 
0 "" 0 "" {TEXT -1 120 "the first three terms will contribute only on \+
the diagonal, and the last two terms only one line away from the diago
nal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "S
uppose we want to couple " }{TEXT 274 2 "J1" }{TEXT -1 9 "=3/2 and " }
{TEXT 273 2 "J2" }{TEXT -1 33 "=1. We work in units where h-bar " }
{TEXT 19 6 "h_ = 1" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "restart; interface(warnlevel=0): with(linalg):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "j1:=3/2; j2:=1;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#j1G#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#j2G\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Normally, w
e are interested in picking an allowed valueof total angular momentum \+
to which " }{TEXT 19 2 "j1" }{TEXT -1 5 " and " }{TEXT 19 2 "j2" }
{TEXT -1 32 " couple (i.e., out of the range " }{TEXT 19 18 "|j1-j2| .
. (j1+j2)" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 56 "However, th
e way things will work in the approach where " }{TEXT 275 1 "J" }
{TEXT -1 26 "^2 is diagonalized in the " }{TEXT 19 13 "|j1 j2 m1 m2>" 
}{TEXT -1 45 " basis, we will not use the specification of " }{TEXT 
19 1 "J" }{TEXT -1 55 ", but rather find the CG coefficients for all p
ossible " }{TEXT 19 1 "J" }{TEXT -1 22 " values. To leave the " }
{TEXT 19 1 "z" }{TEXT -1 21 "-projection of total " }{TEXT 276 1 "J" }
{TEXT -1 69 " as unrestricted as possible we determine the maximum all
owed length:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "J:=j1+j2;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG#\"\"&\"\"#" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 25 "Our task is to represent " }{TEXT 277 1 "J" }
{TEXT -1 10 "^2 in the " }{TEXT 19 13 "|j1 j2 m1 m2>" }{TEXT -1 12 " b
asis with " }{TEXT 19 2 "j1" }{TEXT -1 5 " and " }{TEXT 19 2 "j2" }
{TEXT -1 87 " specified through their quantum numbers. This means that
 the matrix representation of " }{TEXT 278 1 "J" }{TEXT -1 21 "^2 can \+
be labeled by " }{TEXT 19 2 "m1" }{TEXT -1 5 " and " }{TEXT 19 2 "m2" 
}{TEXT -1 11 ". However, " }{TEXT 19 2 "m1" }{TEXT -1 5 " and " }
{TEXT 19 2 "m2" }{TEXT -1 64 " do not vary independently, but are rest
ricted by the condition " }{TEXT 19 11 "m1 + m2 = M" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 53 "Therefore, we now pick a projection out \+
of the range " }{TEXT 19 7 "M=-J..J" }{TEXT -1 163 ". Let us pick anyo
ne but the largest (or lowest) so that we will get some coupling. The \+
reason for avoiding the largest projection is that only one combinatio
n of " }{TEXT 19 1 "z" }{TEXT -1 16 "-projections of " }{TEXT 280 2 "J
1" }{TEXT -1 5 " and " }{TEXT 279 2 "J2" }{TEXT -1 20 " works in that \+
case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "M:=-J+j2+1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"MG#!\"\"\"\"#" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 20 "The solution of the " }{TEXT 290 1 "J" }{TEXT -1 
47 "^2 eigenvalue/eigenvector problem will give us:" }}{PARA 0 "" 0 "
" {TEXT -1 59 "(i) through the eigenvalues the allowed quantum numbers
 of " }{TEXT 259 1 "J" }{TEXT -1 10 "^2, i.e., " }{TEXT 19 12 "h_^2 J \+
(J+1)" }{TEXT -1 31 ", with the possible choices of " }{TEXT 19 1 "J" 
}{TEXT -1 37 " that are compatible with the chosen " }{TEXT 19 1 "M" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 55 "(ii) in anticipation of \+
half-integer spins we pick the " }{TEXT 19 1 "M" }{TEXT -1 45 " value \+
above such that it is compatible with " }{TEXT 19 1 "J" }{TEXT -1 20 "
, i.e,. for integer " }{TEXT 19 1 "J" }{TEXT -1 17 " we pick integer \+
" }{TEXT 19 1 "M" }{TEXT -1 23 ", and for half-integer " }{TEXT 19 1 "
J" }{TEXT -1 22 " we pick half-integer " }{TEXT 19 1 "M" }{TEXT -1 1 "
." }}{PARA 0 "" 0 "" {TEXT -1 34 "(iii) the eigenvector for a given " 
}{TEXT 258 1 "J" }{TEXT -1 125 "^2-eigenvalue gives us the set of coup
ling coefficients that says how the uncoupled states combine to form a
n eigenvector of " }{TEXT 260 1 "J" }{TEXT -1 3 "^2." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Now the question is: ho
w do we find the correct range of " }{TEXT 19 7 "(m1,m2)" }{TEXT -1 
40 " values by use of restrictions (such as " }{TEXT 19 7 "M=m1+m2" }
{TEXT -1 40 ") ? This will determine the matrix size." }}{PARA 0 "" 0 
"" {TEXT -1 66 "We determine by a counting method the matrix size in t
he variable " }{TEXT 19 2 "ic" }{TEXT -1 39 ", and the configurations \+
are stored in " }{TEXT 19 2 "cf" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 186 "ic:=0: for m1 from -j1 to j1 do: for m2 from \+
-j2 to j2 do: if m1+m2=M then ic:=ic+1: cf[ic]:=[m1,m2]: print(`combo \+
`,ic,` involves [m1,m2]= `,cf[ic],` with m1+m2=M: `,M); fi; od: od: ic
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'combo~G\"\"\"%4~involves~[m1,m2
]=~G7$#!\"$\"\"#F$%0~with~m1+m2=M:~G#!\"\"F)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6(%'combo~G\"\"#%4~involves~[m1,m2]=~G7$#!\"\"F$\"\"!%0~w
ith~m1+m2=M:~GF'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%'combo~G\"\"$%4~i
nvolves~[m1,m2]=~G7$#\"\"\"\"\"#!\"\"%0~with~m1+m2=M:~G#F*F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
23 "Now we understand that " }{TEXT 19 2 "j2" }{TEXT -1 59 " (the smal
ler of the two) steps through all allowed values:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "Jsq:=matrix(ic,ic,0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$JsqGK%'matrixG6#7%7%\"\"!F*F*F)F)Q(pprint06\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The matrix is a square matrix: we \+
let the row and column indeces go through all allowed values of " }
{TEXT 19 5 "m1,m2" }{TEXT -1 29 ", i.e., each index goes from " }
{TEXT 19 1 "1" }{TEXT -1 4 " to " }{TEXT 19 2 "ic" }{TEXT -1 15 ". We \+
call them " }{TEXT 19 5 "m1,m2" }{TEXT -1 5 " and " }{TEXT 19 7 "m1p,m
2p" }{TEXT -1 41 " for row and column indices respectively." }}{PARA 
0 "" 0 "" {TEXT -1 66 "We also need indeces to refer to the matrix ent
ries (ranging from " }{TEXT 19 1 "1" }{TEXT -1 4 " to " }{TEXT 19 2 "i
c" }{TEXT -1 9 ") called " }{TEXT 19 3 "im2" }{TEXT -1 5 " and " }
{TEXT 19 4 "im2p" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 128 "To u
nderstand the diagonal matrix entries is straightforward. They are sim
ply the sum of the diagonal elements of the operators " }{TEXT 19 20 "
J1^2+J2^2+ 2*J1z*J2z" }{TEXT -1 46 ", i.e., of the eigenvalues of thes
e operators." }}{PARA 0 "" 0 "" {TEXT -1 76 "The off-diagonal elements
 require one to understand the following structure:" }}{PARA 0 "" 0 "
" {TEXT -1 5 "when " }{TEXT 19 8 "m2=m2p+1" }{TEXT -1 17 " it follows \+
from " }{TEXT 19 15 "m1=M-m2=M-m2p-1" }{TEXT -1 5 " and " }{TEXT 19 
14 "m1p=M-m2p=m1+1" }{TEXT -1 6 " that " }{TEXT 19 8 "m1=m1p-1" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Vice versa: when " }
{TEXT 19 8 "m2=m2p-1" }{TEXT -1 6 " then " }{TEXT 19 8 "m1=m2p+1" }
{TEXT -1 83 ". This allows J1+J2-  and J1-J2+ to have entries just abo
ve and below the diagonal." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "for irow from 1 to ic do: m1:=cf[irow][1]: m2:=cf[irow][2]:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "for icol from 1 to ic do: m1p:=cf[i
col][1]: m2p:=cf[icol][2]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "if m2
=m2p then Jsq[irow,icol]:=j1*(j1+1)+j2*(j2+1)+2*m1*m2: " }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 128 "elif m1=m1p-1 and m2=m2p+1 then Jsq[irow,icol
]:=sqrt((j2-m2+1)*(j2+m2)*(j1+m1+1)*(j1-m1)): print(m1,m1p,m2,m2p,Jsq[
irow,icol]); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "elif m1=m1p+1 and
 m2=m2p-1 then Jsq[irow,icol]:=sqrt((j1-m1+1)*(j1+m1)*(j2+m2+1)*(j2-m2
)): print(m1,m1p,m2,m2p,Jsq[irow,icol]); " }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "fi: od: od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'#!\"$
\"\"##!\"\"F%\"\"\"\"\"!*$\"\"'#F(F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6'#!\"\"\"\"##!\"$F%\"\"!\"\"\"*$\"\"'#F)F%" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6'#!\"\"\"\"##\"\"\"F%\"\"!F$,$*&F%F'F%F&F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6'#\"\"\"\"\"##!\"\"F%F'\"\"!,$*&F%F$F%F#F$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "print(Jsq);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#K%'matrixG6#7%7%#\"#6\"\"%*$\"\"'#\"\"\"\"\"#\"\"!
7%F+#\"#BF*,$*&F/F.F/F-F.7%F0F4#\"#>F*Q(pprint16\"" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 68 "To find out the allowed total angular momentum fo
r which the chosen " }{TEXT 19 1 "M" }{TEXT -1 89 " sublevel can be re
alized we calculate the eigenvalues and keep in mind that they follow \+
" }{TEXT 19 12 "h_^2*J*(J+1)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "eigenvalues(Jsq);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6%#\"\"$\"\"%#\"#:F%#\"#NF%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
40 "Among the three possible eigenvalues of " }{TEXT 282 1 "J" }{TEXT 
-1 60 "^2 we do find the previously selected maximum configuration " }
{TEXT 19 7 "J=j1+j2" }{TEXT -1 25 ", as well as the minimum " }{TEXT 
19 7 "J=j1-j2" }{TEXT -1 21 ", and one in-between." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "J*(J+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
#\"#N\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "abs(j1-j2)*(a
bs(j1-j2)+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"$\"\"%" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "We obtain the mixing coefficients \+
that tell us the following:" }}{PARA 0 "" 0 "" {TEXT -1 38 "the uncoup
led basis states labeled by " }{TEXT 19 7 "[j1 m1]" }{TEXT -1 5 " and \+
" }{TEXT 19 7 "[j2 m2]" }{TEXT -1 19 " can be coupled to " }{TEXT 19 
5 "[J M]" }{TEXT -1 27 " values specified by fixed " }{TEXT 19 1 "M" }
{TEXT -1 19 " and by calculated " }{TEXT 19 1 "J" }{TEXT -1 30 " (from
 the diagonalization of " }{TEXT 283 1 "J" }{TEXT -1 101 "^2). The mea
ning of the eigenvector entries is that the coeffcients tell how much \+
of the product of  " }{TEXT 19 7 "[j1 m1]" }{TEXT -1 5 " and " }{TEXT 
19 7 "[j2 m2]" }{TEXT -1 23 " states is needed with " }{TEXT 19 5 "m1,
m2" }{TEXT -1 37 " given by the specification in table " }{TEXT 19 2 "
cf" }{TEXT -1 75 ". Thus the first eigenvector component is the admixi
ng coefficient for the " }{TEXT 19 8 "(m1, m2)" }{TEXT -1 17 " values \+
given in " }{TEXT 19 5 "cf[1]" }{TEXT -1 61 ", the second eigenvector \+
entry is the one for the product of " }{TEXT 19 7 "[j1 m1]" }{TEXT -1 
6 " with " }{TEXT 19 7 "[j2 m2]" }{TEXT -1 6 " with " }{TEXT 19 2 "m1
" }{TEXT -1 5 " and " }{TEXT 19 2 "m2" }{TEXT -1 10 " given in " }
{TEXT 19 5 "cf[2]" }{TEXT -1 6 ", etc." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "vec:=eigenvects(Jsq);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$vecG6%7%#\"#:\"\"%\"\"\"<#K%'vect
orG6#7%*$\"\"'#F*\"\"#F*,$*&F3F*F3F2!\"\"Q(pprint26\"7%#\"\"$F)F*<#KF-
6#7%,$*&F3F6F1F2F6F*,$*&F3F6F3F2F6Q(pprint3F87%#\"#NF)F*<#KF-6#7%,$*&F
1F6F1F2F*F*,$*&F3F6F3F2F*Q(pprint4F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 53 "The eigenvector corresponding to the chosen value of " }{TEXT 
19 1 "J" }{TEXT -1 49 " when normalized gives the coupling coefficient
s." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "nops([vec]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 130 "for i from 1 to nops([vec]) do: if vec[i][1]=J*(J+1)
 then myvec:=op(vec[i][3]); print(`unnormalized eigenvector: `,myvec);
 fi: od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;unnormalized~eigenvector
:~GK%'vectorG6#7%,$*&\"\"'!\"\"F*#\"\"\"\"\"#F-F-,$*&F.F+F.F,F-Q(pprin
t56\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "CG:=map(combine,no
rmalize(myvec));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#CGGK%'vectorG6#
7%,$*&\"#I!\"\"\"#!*#\"\"\"\"\"#F/,$*&\"\"&F,\"#:F.F/,$*&\"#5F,F+F.F/Q
(pprint66\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "The normalization
 is important to maintain unitarity, and thus a probabilistic interpre
tation. If we prepare the system in a " }{TEXT 19 14 "[J, M, j1, j2]" 
}{TEXT -1 30 " eigenstate, and then measure " }{TEXT 19 4 "j1_z" }
{TEXT -1 5 " and " }{TEXT 19 4 "j2_z" }{TEXT -1 113 ", then the square
 of the CG coefficient will provide us the probabilities to measure a \+
particular set of allowed " }{TEXT 19 8 "(m1, m2)" }{TEXT -1 24 " valu
es. First we check:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "add(
CG[i]^2,i=1..vectdim(CG));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(cf);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/\"\"\"7$#!\"$\"\"#F(/F,7$#!\"\"F,
\"\"!/\"\"$7$#F(F,F0Q(pprint76\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "CG[1]^2,CG[2]^2,CG[3]^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%#\"\"\"\"#5#\"\"$\"\"&#F'F%" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 41 "The statement is that when we are in the " }{TEXT 19 30 "
|J=5/2, M=-1/2, j1=3/2, j2=1 >" }{TEXT -1 40 " eigenstate, then the pr
obabilities are:" }}{PARA 0 "" 0 "" {TEXT -1 17 "10% to be in the " }
{TEXT 19 19 "(m1,m2) = (-3/2, 1)" }{TEXT -1 12 " combination" }}{PARA 
0 "" 0 "" {TEXT -1 17 "60% to be in the " }{TEXT 19 19 "(m1,m2) = (-1/
2, 0)" }{TEXT -1 12 " combination" }}{PARA 0 "" 0 "" {TEXT -1 17 "30% \+
to be in the " }{TEXT 19 19 "(m1,m2) = (1/2, -1)" }{TEXT -1 13 " combi
nation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 285 11 "Exercise1 :" }}{PARA 0 "" 0 "" {TEXT -1 49 "Go through t
he calculation for the other allowed " }{TEXT 19 1 "M" }{TEXT -1 21 " \+
values and the same " }{TEXT 19 1 "J" }{TEXT -1 14 " value, i.e., " }
{TEXT 19 7 "J=j1+j2" }{TEXT -1 65 ". Calculate the probabilities to fi
nd the system in a particular " }{TEXT 19 7 "(m1,m2)" }{TEXT -1 60 " c
onfiguration and display graphically using arrows for the " }{TEXT 19 
7 "(j1,m1)" }{TEXT -1 5 " and " }{TEXT 19 7 "(j2,m2)" }{TEXT -1 34 " a
ngular momentum vectors how the " }{TEXT 19 5 "(J M)" }{TEXT -1 19 " s
tate comes about." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 286 11 "Exercise2 :" }}{PARA 0 "" 0 "" {TEXT -1 18 "Look at the \+
other " }{TEXT 19 1 "J" }{TEXT -1 29 " values for which the set of " }
{TEXT 19 7 "(j1 m1)" }{TEXT -1 5 " and " }{TEXT 19 9 "(j2 M-m1)" }
{TEXT -1 27 " configurations couples to " }{TEXT 19 5 "(J M)" }{TEXT 
-1 98 " and construct the probabilities as well as the graphical repre
sentations discussed in exercise 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 287 11 "Exercise 3:" }}{PARA 0 "" 0 "" {TEXT 
-1 28 "Pick another combination of " }{TEXT 19 2 "j1" }{TEXT -1 5 " an
d " }{TEXT 19 2 "j2" }{TEXT -1 52 ", and repeat the calculations for e
xercises 1 and 2." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 284 41 "A program for Clebsch-Gordan coefficients" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "One can a
rrive at a general formula given, e.g., in William J. Thompson: " }
{TEXT 257 16 "Angular Momentum" }{TEXT -1 141 ", Wiley 1994, eq. 7.51.
 One introduces first the related (3j) coefficient for coupling three \+
angular momenta to zero. It involves a sum over " }{TEXT 19 1 "k" }
{TEXT -1 37 " (which is carried out as a sum over " }{TEXT 19 3 "2*k" 
}{TEXT -1 52 " to allow half-integer angular momentum projection)." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "ThreeJ:=proc(j1,m1,j2,m2,j3
,m3) local k,tk,tkmin,tkmax,res,phas,n1,n2,d1,d2,d3,term;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "if m1+m2+m3 <> 0 then RETURN(0) fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "tkmin:=2*max(0,j2-j1-m3);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "tkmax:=2*min(j3-m3,j3-j1+j2); res:=0;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "phas:=(-1)^(tkmin/2+j2+m2);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for tk from tkmin to tkmax by 2 do:
 k:=tk/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "n1:=(j2+j3+m1-k)!; #pr
int(\"n1= \",n1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "n2:=(j1-m1+k)!
; #print(\"n2= \",n2);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "d1:=k!*(j
3-j1+j2-k)!; #print(\"d1= \",d1);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "d2:=(j3-m3-k)!; #print(\"d2= \",d2,k,j1,j2,m3);" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 37 "d3:=(k+j1-j2+m3)!; #print(\"d3= \",d3);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 28 "term:=phas*n1*n2/(d1*d2*d3);" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "phas:=-phas;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "res:=res+term; od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "simp
lify(res)*(-1)^(j1-j2-m3)*sqrt((j3+j1-j2)!*(j3-j1+j2)!*(j1+j2-j3)!*(j3
-m3)!*(j3+m3)!/((j1+j2+j3+1)!*(j1-m1)!*(j1+m1)!*(j2-m2)!*(j2+m2)!)); e
nd:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "CGC:=(j1,m1,j2,m2,J)
->combine(ThreeJ(j1,m1,j2,m2,J,-m1-m2)*sqrt(2*J+1)*(-1)^(j1-j2+m1+m2))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$CGCGf*6'%#j1G%#m1G%#j2G%#m2G%
\"JG6\"6$%)operatorG%&arrowGF,-%(combineG6#*(-%'ThreeJG6(9$9%9&9'9(,&F
8!\"\"F:F=\"\"\"-%%sqrtG6#,&*&\"\"#F>F;F>F>F>F>F>)F=,*F7F>F9F=F8F>F:F>
F>F,F,F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "For our example we ha
ve " }{TEXT 19 5 "j1 j2" }{TEXT -1 18 " and the range of " }{TEXT 19 
2 "m2" }{TEXT -1 5 " and " }{TEXT 19 2 "m1" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "j1,j2,[-1,0,1],[M-(-1),M,M-1]; #the
 matrix was done by stepping with m2 [-1,0,1]." }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6&#\"\"$\"\"#\"\"\"7%!\"\"\"\"!F&7%#F&F%#F(F%#!\"$F%" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "CG[1];" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&\"#I!\"\"\"#!*#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "CGC(j1,M+1,j2,-1,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$*&\"#5!\"\"\"#I#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "%-%%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5!\"\"
\"#I#\"\"\"\"\"#F)*&F'F&\"#!*F(F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "CG[2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"&!\"
\"\"#:#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "CGC
(j1,M,j2,0,J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#5!\"\"\"#g#\"
\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "simplify(%-%
%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "CGC(j1,M-1,j2,1,J);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&\"#I!\"\"\"#!*#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "CG[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&
\"#5!\"\"\"#I#\"\"\"\"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "%-%%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"#5!\"\"\"#I#\"\"\"\"
\"#F)*&F'F&\"#!*F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "In a nuts
hell the meaning of the calculation is:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT 19 81 "|JM j1 j2> = |5/2, -1/2, 3/2, 1> = a
dd( CG[M-m2,m2]* |j1,M-m2> |j2 m2> ,m2=-1..1)" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "J,M,j1,j2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&#
\"\"&\"\"##!\"\"F%#\"\"$F%\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
170 "There are sum rules for CG-coefficients given in Thompson's book \+
(or other books on group theory/angular momentum algebra), which can b
e checked by explicit calculations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 288 11 "Application" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "One useful application of CG c
oefficients or (3j) symbols is in the evaluation of angular integrals \+
over three spherical harmonics. W.J. Thompson lists it as eq. (7.102) \+
and shows how it follows as a special case of an integral over three W
igner rotation matrices. We start with a definition of the spherical h
armonics." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "with(orthopoly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 53 "Plm:=proc(theta,l::nonnegint,m::integer) local x,y,f;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "x:=cos(theta);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 42 "if m>0 then f:=subs(y=x,diff(P(l,y),y$m));" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "else f:=subs(y=x,P(l,y)); fi;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "(-1)^m*sin(theta)^m*f; end:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Plm(theta,3,2);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*(\"#:\"\"\")-%$sinG6#%&thetaG\"\"#F&-%$cosGF
*F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "For the spherical harmon
ics we don't need the Plm's with negative argument." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "Y:=proc(theta,phi,l::nonnegint,m::integer
) local m1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "m1:=abs(m); if m1>l \+
then RETURN(\"|m\} has to be <= l for Y_lm\"); fi;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 78 "exp(I*m*phi)*Plm(theta,l,m1)*(-1)^m*sqrt((2*l+1)*(l
-m1)!/(4*Pi*(l+m1)!)); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "Y(theta,phi,3,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"
\"#F&*(,&*&#\"\"&F'F&*$)-%$cosG6#%&thetaG\"\"$F&F&F&*&#F3F'F&F/F&!\"\"
F&\"\"(F%%#PiG#F6F'F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
Y(theta,phi,3,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"#7F&
*,-%$expG6#*&%$phiGF&^#F&F&F&-%$sinG6#%&thetaGF&,&*&#\"#:\"\"#F&*$)-%$
cosGF1F7F&F&F&#\"\"$F7!\"\"F&\"#@#F&F7%#PiG#F>F7F&F&" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 108 "No:=(l,m)->int(int(evalc(Y(theta,phi,l,m
)*conjugate(Y(theta,phi,l,m))),phi=0..2*Pi)*sin(theta),theta=0..Pi);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NoGf*6$%\"lG%\"mG6\"6$%)operatorG
%&arrowGF)-%$intG6$*&-F.6$-%&evalcG6#*&-%\"YG6&%&thetaG%$phiG9$9%\"\"
\"-%*conjugateG6#F7F>/F;;\"\"!,$*&\"\"#F>%#PiGF>F>F>-%$sinG6#F:F>/F:;F
DFHF)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "No(1,0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "Now suppose we are interested in the following integral:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "AI:=(J,M,j1,m1,j2,m2)-
>int(int(evalc(Y(theta,phi,j1,m1)*Y(theta,phi,j2,m2)*conjugate(Y(theta
,phi,J,M))),phi=0..2*Pi)*sin(theta),theta=0..Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#AIGf*6(%\"JG%\"MG%#j1G%#m1G%#j2G%#m2G6\"6$%)operator
G%&arrowGF--%$intG6$*&-F26$-%&evalcG6#*(-%\"YG6&%&thetaG%$phiG9&9'\"\"
\"-F<6&F>F?9(9)FB-%*conjugateG6#-F<6&F>F?9$9%FB/F?;\"\"!,$*&\"\"#FB%#P
iGFBFBFB-%$sinG6#F>FB/F>;FPFTF-F-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 80 "There are restrictions on the angular momentum parameters: (i) \+
triangle rule on " }{TEXT 19 9 "(j1,j2,J)" }{TEXT -1 7 "; (ii) " }
{TEXT 19 7 "j1+j2+J" }{TEXT -1 23 " = even integer; (iii) " }{TEXT 19 
7 "m1+m2=M" }{TEXT -1 14 ". For example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "AI(2,1,1,0,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,
$**\"#5!\"\"\"\"$#\"\"\"\"\"#%#PiG#F&F*\"\"&F(F)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 59 "In terms of (3j) symbols the integral can be calcu
lated as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "AI3j:=(J,M,j1
,m1,j2,m2)->simplify((-1)^M*sqrt((2*j1+1)*(2*j2+1)*(2*J+1)/(4*Pi))*Thr
eeJ(j1,m1,j2,m2,J,-M)*ThreeJ(j1,0,j2,0,J,0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%AI3jGf*6(%\"JG%\"MG%#j1G%#m1G%#j2G%#m2G6\"6$%)operat
orG%&arrowGF--%)simplifyG6#**)!\"\"9%\"\"\"-%%sqrtG6#,$*&#F8\"\"%F8**,
&*&\"\"#F89&F8F8F8F8F8,&*&FCF89(F8F8F8F8F8,&*&FCF89$F8F8F8F8F8%#PiGF6F
8F8F8-%'ThreeJG6(FD9'FG9)FJ,$F7F6F8-FM6(FD\"\"!FGFTFJFTF8F-F-F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "AI3j(2,1,1,0,1,1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#,$**\"#5!\"\"\"\"$#\"\"\"\"\"#%#PiG#F&F*\"\"
&F(F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 289 11 "Exercise 4:" }}{PARA 0 
"" 0 "" {TEXT -1 187 "Verify the short-cut formula by comparing it aga
inst the symbolic computation of the integral. Go to large values of t
he angular momentum parameters and check its computational advantage.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 294 "Integrals of this type arise in atomic p
hysics in the computation of multipole expansions of the potential of \+
a charge distribution, and in the computation of dipole matrix element
s (e.g., in the radiative decay) in which the Cartesian components of \+
the position vector are represented through " }{TEXT 19 8 "r*Y(1,m)" }
{TEXT -1 8 ", e.g., " }{TEXT 19 14 "x+I*y=r*Y(1,1)" }{TEXT -1 1 "." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 2 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }