22.2 Evaluation
255
22.1.4 Wigner 9j-Symbols
Similar to the 6j-symbol the 9j-symbol is proportional to the recoupling matrix
for four angular momenta. For four angular momenta j 1 , j 2 , j 3 , j 4 we have the
following possibilities: (j 1 , j 2 ) → j 12 ; (j 3 , j 4 ) → j 34 ; (j 12 , j 34 ) → J and
(j 1 , j 3 ) → j 13 ; (j 2 , j 4 ) → j 24 ; (j 13 , j 24 ) → J . The corresponding 9j-symbol
is given by Edmonds [2]
⎧
⎨
⎩
j 1 j 2 j 12
j 3 j 4 j 34
j 13 j 24 J
⎫
⎬
⎭
and can be evaluated via [3]
⎧
⎨
⎩
j 11 j 12 j 13
j 21 j 22 j 23
j 31 j 32 j 33
⎫
⎬
⎭
=
m 11 ,...,m 33
j 11 j 12 j 13
m 11 m 12 m 13
j 21 j 22 j 23
m 21 m 22 m 23
×
j 31 j 32 j 33
m 31 m 32 m 33
j 11 j 21 j 31
m 11 m 21 m 31
×
j 12 j 22 j 32
m 12 m 22 m 32
j 13 j 23 j 33
m 13 m 23 m 33
,
(22.16)
where the summation is about all allowed quantum number m ij .
22.2 Evaluation
22.2.1 Clebsch-Gordan Coefficients and Wigner 3j-symbols
Clebsch-Gordan Coefficients
The Clebsch-Gordan coefficients can be evaluated with SPECFUNPHYS class
jjCG. The syntax is [obj, er] = jjCG(j1,j2,j3,m1,m2,m3) with
j 3 = J and m3 = M = m 1 + m 2 , see Eq. (22.10). The computation is based on
Eqs. (22.12) and (22.14). The output arguments are the class object “obj” with a
table (property Clebsch-Gordan) of all relevant Clebsch-Gordan coefficients and
some general information and a vector or array “er” with the same information as
obj.ClebschGordan. Syntax: [obj, er] = jjCG(j1,j2) all allowed
Clebsch-Gordan coefficients based on the angular momenta j 1 , j 2 will be evaluated.
With [obj, er] = jjCG(j1,j2,m3) the total list will be restricted on
Clebsch-Gordan coefficients with value M = m 3 , and additionally restricted on the
total angular momentum j 3 via [obj, er] = jjCG(j1,j2,j3,m3).
[obj, er] = jjCG(j1,j2,j3,m1,m2) or
[obj, er] = jjCG(j1,j2,j3,m1,m2,m3) evaluates the single ClebschGordan coefficient with m 1 + m 2 = m 3 .