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22 Wigner- and Clebsch-Gordan Coefficients
Wigner 3j-Symbols The SPECFUNPHYS class jj3j returns the values of
the Wigner 3j-symbol. The evaluation is based on Eq. (22.14). (Please note:
m 1 + m 2 + m 3 = 0.) The syntax follows the class jjCG: [obj, er] =
jj3j(j1,j2); all allowed values based on the angular momenta j 1 , j 2 will
be evaluated.
[obj, er] = jj3j(j1,j2,m3); the evaluation will be restricted to magnetic
quantum number equal to m 3
[obj, er] = jj3j(j1,j2,j3,m3); the evaluation will be in addition
restricted to angular momentum j 3 , and finally with
[obj, er] = jj3j(j1,j2,j3,m1,m2) or
[obj, er] = jj3j(j1,j2,j3,m1,m2,m3)
only the single value based on all quantum numbers will be computed. The output
variable is again the object “obj” with properties “JJ3j”, a table of the corresponding
values of the 3j-symbol and “info” with some additional information. “er” is an array
with the values of the 3j-symbol.
Example
>> obj = jj3j(0.5,1,0.5);
% j1=0.5, j2=1, m3=0.5
>> obj.JJ3j
ans =
4x7 table
j1
j2
j3
m1
m2
m3
jj3j
___
__
___
____
__
___
________
0.5
1
0.5
-0.5
0
0.5
0.40825
0.5
1
0.5
0.5
-1
0.5
-0.57735
0.5
1
1.5
-0.5
0
0.5
-0.40825
0.5
1
1.5
0.5
-1
0.5
-0.28868
22.2.2 Wigner 6j-Symbols
The evaluation of the Wigner 6j-symbols is based on Eq. (22.15). The syntax of
the SPECFUNPHYS class jj6j is [obj, er] = jj6J(j1,j2,j3) for the
evaluation of all allowed values for the three angular momenta j 1 , j 2 , j 3 we want to
couple and [obj, er] = jj6j(j1,j2,j3,J) restricted to the total angular
momentum J . With [obj, er] = jj6j(j1,j2,j2,j3,j4,j5,j6) the
corresponding single value will be computed.
Example
>> obj = jj6j(1,0.5,2,2.5); % j1=1, j2=0.5, j3=2, J=2.5
>> obj.JJ6j
ans =
4x7 table