From Wikipedia, the free encyclopedia
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics . The overall sign of the coefficients for each set of constant
j
1
{\displaystyle j_{1}}
,
j
2
{\displaystyle j_{2}}
,
j
{\displaystyle j}
is arbitrary to some degree and has been fixed according to the Condon-Shortley and Wigner sign convention as discussed by Baird and Biedenharn .[1] Tables with the same sign convention may be found in the Particle Data Group 's Review of Particle Properties [2] and in online tables.[3]
Formulation
The Clebsch–Gordan coefficients are the solutions to
|
(
j
1
j
2
)
j
m
⟩
=
∑
m
1
=
−
j
1
j
1
∑
m
2
=
−
j
2
j
2
|
j
1
m
1
j
2
m
2
⟩
⟨
j
1
j
2
;
m
1
m
2
|
j
1
j
2
;
j
m
⟩
{\displaystyle |(j_{1}j_{2})jm\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle }
Explicitly:
⟨
j
1
j
2
;
m
1
m
2
|
j
1
j
2
;
j
m
⟩
=
δ
m
,
m
1
+
m
2
(
2
j
+
1
)
(
j
+
j
1
−
j
2
)
!
(
j
−
j
1
+
j
2
)
!
(
j
1
+
j
2
−
j
)
!
(
j
1
+
j
2
+
j
+
1
)
!
×
(
j
+
m
)
!
(
j
−
m
)
!
(
j
1
−
m
1
)
!
(
j
1
+
m
1
)
!
(
j
2
−
m
2
)
!
(
j
2
+
m
2
)
!
×
∑
k
(
−
1
)
k
k
!
(
j
1
+
j
2
−
j
−
k
)
!
(
j
1
−
m
1
−
k
)
!
(
j
2
+
m
2
−
k
)
!
(
j
−
j
2
+
m
1
+
k
)
!
(
j
−
j
1
−
m
2
+
k
)
!
.
{\displaystyle {\begin{aligned}\langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =\ &\delta _{m,m_{1}+m_{2}}{\sqrt {\frac {(2j+1)(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!(j_{1}+j_{2}-j)!}{(j_{1}+j_{2}+j+1)!}}}\ \times \\&{\sqrt {(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times \\&\sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j-j_{2}+m_{1}+k)!(j-j_{1}-m_{2}+k)!}}.\end{aligned}}}
The summation is extended over all integer k for which the argument of every factorial is nonnegative.[4]
For brevity, solutions with m < 0 and j 1 < j 2 are omitted. They may be calculated using the simple relations
⟨
j
1
j
2
;
m
1
m
2
|
j
1
j
2
;
j
m
⟩
=
(
−
1
)
j
−
j
1
−
j
2
⟨
j
1
j
2
;
−
m
1
,
−
m
2
|
j
1
j
2
;
j
,
−
m
⟩
{\displaystyle \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{1}j_{2};-m_{1},-m_{2}|j_{1}j_{2};j,-m\rangle }
.
and
⟨
j
1
j
2
;
m
1
m
2
|
j
1
j
2
;
j
m
⟩
=
(
−
1
)
j
−
j
1
−
j
2
⟨
j
2
j
1
;
m
2
m
1
|
j
2
j
1
;
j
m
⟩
{\displaystyle \langle j_{1}j_{2};m_{1}m_{2}|j_{1}j_{2};jm\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{2}j_{1};m_{2}m_{1}|j_{2}j_{1};jm\rangle }
.
A complete list [5]
j 2 =0
When j 2 = 0 , the Clebsch–Gordan coefficients are given by
δ
j
,
j
1
δ
m
,
m
1
{\displaystyle \delta _{j,j_{1}}\delta _{m,m_{1}}}
.
j 1 =1 / 2 , j 2 =1 / 2
m =1
j
m 1 , m 2
1
1 / 2 , 1 / 2
1
{\displaystyle 1\!\,}
m =-1
j
m 1 , m 2
1
-1 / 2 , -1 / 2
1
{\displaystyle 1\!\,}
m =0
j
m 1 , m 2
1
0
1 / 2 , -1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
-1 / 2 , 1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
j 1 =1, j 2 =1 / 2
m =3 / 2
j
m 1 , m 2
3 / 2
1, 1 / 2
1
{\displaystyle 1\!\,}
m =1 / 2
j
m 1 , m 2
3 / 2
1 / 2
1, -1 / 2
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
0, 1 / 2
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
−
1
3
{\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}
j 1 =1, j 2 =1
m =2
j
m 1 , m 2
2
1, 1
1
{\displaystyle 1\!\,}
m =1
j
m 1 , m 2
2
1
1, 0
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
0, 1
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
m =0
j
m 1 , m 2
2
1
0
1, -1
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
0, 0
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
0
{\displaystyle 0\!\,}
−
1
3
{\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}
-1, 1
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
j 1 =3 / 2 , j 2 =1 / 2
m =2
j
m 1 , m 2
2
3 / 2 , 1 / 2
1
{\displaystyle 1\!\,}
m =1
j
m 1 , m 2
2
1
3 / 2 , -1 / 2
1
2
{\displaystyle {\frac {1}{2}}\!\,}
3
4
{\displaystyle {\sqrt {\frac {3}{4}}}\!\,}
1 / 2 , 1 / 2
3
4
{\displaystyle {\sqrt {\frac {3}{4}}}\!\,}
−
1
2
{\displaystyle -{\frac {1}{2}}\!\,}
m =0
j
m 1 , m 2
2
1
1 / 2 , -1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
-1 / 2 , 1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
j 1 =3 / 2 , j 2 =1
m =5 / 2
j
m 1 , m 2
5 / 2
3 / 2 , 1
1
{\displaystyle 1\!\,}
m =3 / 2
j
m 1 , m 2
5 / 2
3 / 2
3 / 2 , 0
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
1 / 2 , 1
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
m =1 / 2
j
m 1 , m 2
5 / 2
3 / 2
1 / 2
3 / 2 , -1
1
10
{\displaystyle {\sqrt {\frac {1}{10}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1 / 2 , 0
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
1
15
{\displaystyle {\sqrt {\frac {1}{15}}}\!\,}
−
1
3
{\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}
-1 / 2 , 1
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
−
8
15
{\displaystyle -{\sqrt {\frac {8}{15}}}\!\,}
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
j 1 =3 / 2 , j 2 =3 / 2
m =3
j
m 1 , m 2
3
3 / 2 , 3 / 2
1
{\displaystyle 1\!\,}
m =2
j
m 1 , m 2
3
2
3 / 2 , 1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1 / 2 , 3 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
m =1
j
m 1 , m 2
3
2
1
3 / 2 , -1 / 2
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
1 / 2 , 1 / 2
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
0
{\displaystyle 0\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
-1 / 2 , 3 / 2
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
m =0
j
m 1 , m 2
3
2
1
0
3 / 2 , -3 / 2
1
20
{\displaystyle {\sqrt {\frac {1}{20}}}\!\,}
1
2
{\displaystyle {\frac {1}{2}}\!\,}
9
20
{\displaystyle {\sqrt {\frac {9}{20}}}\!\,}
1
2
{\displaystyle {\frac {1}{2}}\!\,}
1 / 2 , -1 / 2
9
20
{\displaystyle {\sqrt {\frac {9}{20}}}\!\,}
1
2
{\displaystyle {\frac {1}{2}}\!\,}
−
1
20
{\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}
−
1
2
{\displaystyle -{\frac {1}{2}}\!\,}
-1 / 2 , 1 / 2
9
20
{\displaystyle {\sqrt {\frac {9}{20}}}\!\,}
−
1
2
{\displaystyle -{\frac {1}{2}}\!\,}
−
1
20
{\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}
1
2
{\displaystyle {\frac {1}{2}}\!\,}
-3 / 2 , 3 / 2
1
20
{\displaystyle {\sqrt {\frac {1}{20}}}\!\,}
−
1
2
{\displaystyle -{\frac {1}{2}}\!\,}
9
20
{\displaystyle {\sqrt {\frac {9}{20}}}\!\,}
−
1
2
{\displaystyle -{\frac {1}{2}}\!\,}
j 1 =2, j 2 =1 / 2
m =5 / 2
j
m 1 , m 2
5 / 2
2, 1 / 2
1
{\displaystyle 1\!\,}
m =3 / 2
j
m 1 , m 2
5 / 2
3 / 2
2, -1 / 2
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
4
5
{\displaystyle {\sqrt {\frac {4}{5}}}\!\,}
1, 1 / 2
4
5
{\displaystyle {\sqrt {\frac {4}{5}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
m =1 / 2
j
m 1 , m 2
5 / 2
3 / 2
1, -1 / 2
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
0, 1 / 2
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
j 1 =2, j 2 =1
m =3
j
m 1 , m 2
3
2, 1
1
{\displaystyle 1\!\,}
m =2
j
m 1 , m 2
3
2
2, 0
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
1, 1
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
−
1
3
{\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}
m =1
j
m 1 , m 2
3
2
1
2, -1
1
15
{\displaystyle {\sqrt {\frac {1}{15}}}\!\,}
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
1, 0
8
15
{\displaystyle {\sqrt {\frac {8}{15}}}\!\,}
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
0, 1
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
1
10
{\displaystyle {\sqrt {\frac {1}{10}}}\!\,}
m =0
j
m 1 , m 2
3
2
1
1, -1
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
0, 0
3
5
{\displaystyle {\sqrt {\frac {3}{5}}}\!\,}
0
{\displaystyle 0\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
-1, 1
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
j 1 =2, j 2 =3 / 2
m =7 / 2
j
m 1 , m 2
7 / 2
2, 3 / 2
1
{\displaystyle 1\!\,}
m =5 / 2
j
m 1 , m 2
7 / 2
5 / 2
2, 1 / 2
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
4
7
{\displaystyle {\sqrt {\frac {4}{7}}}\!\,}
1, 3 / 2
4
7
{\displaystyle {\sqrt {\frac {4}{7}}}\!\,}
−
3
7
{\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}
m =3 / 2
j
m 1 , m 2
7 / 2
5 / 2
3 / 2
2, -1 / 2
1
7
{\displaystyle {\sqrt {\frac {1}{7}}}\!\,}
16
35
{\displaystyle {\sqrt {\frac {16}{35}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1, 1 / 2
4
7
{\displaystyle {\sqrt {\frac {4}{7}}}\!\,}
1
35
{\displaystyle {\sqrt {\frac {1}{35}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
0, 3 / 2
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
−
18
35
{\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
m =1 / 2
j
m 1 , m 2
7 / 2
5 / 2
3 / 2
1 / 2
2, -3 / 2
1
35
{\displaystyle {\sqrt {\frac {1}{35}}}\!\,}
6
35
{\displaystyle {\sqrt {\frac {6}{35}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1, -1 / 2
12
35
{\displaystyle {\sqrt {\frac {12}{35}}}\!\,}
5
14
{\displaystyle {\sqrt {\frac {5}{14}}}\!\,}
0
{\displaystyle 0\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
0, 1 / 2
18
35
{\displaystyle {\sqrt {\frac {18}{35}}}\!\,}
−
3
35
{\displaystyle -{\sqrt {\frac {3}{35}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
-1, 3 / 2
4
35
{\displaystyle {\sqrt {\frac {4}{35}}}\!\,}
−
27
70
{\displaystyle -{\sqrt {\frac {27}{70}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
−
1
10
{\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}
j 1 =2, j 2 =2
m =4
j
m 1 , m 2
4
2, 2
1
{\displaystyle 1\!\,}
m =3
j
m 1 , m 2
4
3
2, 1
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1, 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
m =2
j
m 1 , m 2
4
3
2
2, 0
3
14
{\displaystyle {\sqrt {\frac {3}{14}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
1, 1
4
7
{\displaystyle {\sqrt {\frac {4}{7}}}\!\,}
0
{\displaystyle 0\!\,}
−
3
7
{\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}
0, 2
3
14
{\displaystyle {\sqrt {\frac {3}{14}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
m =1
j
m 1 , m 2
4
3
2
1
2, -1
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
1, 0
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
−
1
14
{\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
0, 1
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
−
1
14
{\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
-1, 2
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
m =0
j
m 1 , m 2
4
3
2
1
0
2, -2
1
70
{\displaystyle {\sqrt {\frac {1}{70}}}\!\,}
1
10
{\displaystyle {\sqrt {\frac {1}{10}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
1, -1
8
35
{\displaystyle {\sqrt {\frac {8}{35}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
−
1
10
{\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
0, 0
18
35
{\displaystyle {\sqrt {\frac {18}{35}}}\!\,}
0
{\displaystyle 0\!\,}
−
2
7
{\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}
0
{\displaystyle 0\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
-1, 1
8
35
{\displaystyle {\sqrt {\frac {8}{35}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
1
10
{\displaystyle {\sqrt {\frac {1}{10}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
-2, 2
1
70
{\displaystyle {\sqrt {\frac {1}{70}}}\!\,}
−
1
10
{\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
j 1 =5 / 2 , j 2 =1 / 2
m =3
j
m 1 , m 2
3
5 / 2 , 1 / 2
1
{\displaystyle 1\!\,}
m =2
j
m 1 , m 2
3
2
5 / 2 , -1 / 2
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
5
6
{\displaystyle {\sqrt {\frac {5}{6}}}\!\,}
3 / 2 , 1 / 2
5
6
{\displaystyle {\sqrt {\frac {5}{6}}}\!\,}
−
1
6
{\displaystyle -{\sqrt {\frac {1}{6}}}\!\,}
m =1
j
m 1 , m 2
3
2
3 / 2 , -1 / 2
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
1 / 2 , 1 / 2
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
−
1
3
{\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}
m =0
j
m 1 , m 2
3
2
1 / 2 , -1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
-1 / 2 , 1 / 2
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
j 1 =5 / 2 , j 2 =1
m =7 / 2
j
m 1 , m 2
7 / 2
5 / 2 , 1
1
{\displaystyle 1\!\,}
m =5 / 2
j
m 1 , m 2
7 / 2
5 / 2
5 / 2 , 0
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
5
7
{\displaystyle {\sqrt {\frac {5}{7}}}\!\,}
3 / 2 , 1
5
7
{\displaystyle {\sqrt {\frac {5}{7}}}\!\,}
−
2
7
{\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}
m =3 / 2
j
m 1 , m 2
7 / 2
5 / 2
3 / 2
5 / 2 , -1
1
21
{\displaystyle {\sqrt {\frac {1}{21}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
2
3
{\displaystyle {\sqrt {\frac {2}{3}}}\!\,}
3 / 2 , 0
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
9
35
{\displaystyle {\sqrt {\frac {9}{35}}}\!\,}
−
4
15
{\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}
1 / 2 , 1
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
−
16
35
{\displaystyle -{\sqrt {\frac {16}{35}}}\!\,}
1
15
{\displaystyle {\sqrt {\frac {1}{15}}}\!\,}
m =1 / 2
j
m 1 , m 2
7 / 2
5 / 2
3 / 2
3 / 2 , -1
1
7
{\displaystyle {\sqrt {\frac {1}{7}}}\!\,}
16
35
{\displaystyle {\sqrt {\frac {16}{35}}}\!\,}
2
5
{\displaystyle {\sqrt {\frac {2}{5}}}\!\,}
1 / 2 , 0
4
7
{\displaystyle {\sqrt {\frac {4}{7}}}\!\,}
1
35
{\displaystyle {\sqrt {\frac {1}{35}}}\!\,}
−
2
5
{\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}
-1 / 2 , 1
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
−
18
35
{\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
j 1 =5 / 2 , j 2 =3 / 2
m =4
j
m 1 , m 2
4
5 / 2 , 3 / 2
1
{\displaystyle 1\!\,}
m =3
j
m 1 , m 2
4
3
5 / 2 , 1 / 2
3
8
{\displaystyle {\sqrt {\frac {3}{8}}}\!\,}
5
8
{\displaystyle {\sqrt {\frac {5}{8}}}\!\,}
3 / 2 , 3 / 2
5
8
{\displaystyle {\sqrt {\frac {5}{8}}}\!\,}
−
3
8
{\displaystyle -{\sqrt {\frac {3}{8}}}\!\,}
m =2
j
m 1 , m 2
4
3
2
5 / 2 , -1 / 2
3
28
{\displaystyle {\sqrt {\frac {3}{28}}}\!\,}
5
12
{\displaystyle {\sqrt {\frac {5}{12}}}\!\,}
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
3 / 2 , 1 / 2
15
28
{\displaystyle {\sqrt {\frac {15}{28}}}\!\,}
1
12
{\displaystyle {\sqrt {\frac {1}{12}}}\!\,}
−
8
21
{\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}
1 / 2 , 3 / 2
5
14
{\displaystyle {\sqrt {\frac {5}{14}}}\!\,}
−
1
2
{\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}
1
7
{\displaystyle {\sqrt {\frac {1}{7}}}\!\,}
m =1
j
m 1 , m 2
4
3
2
1
5 / 2 , -3 / 2
1
56
{\displaystyle {\sqrt {\frac {1}{56}}}\!\,}
1
8
{\displaystyle {\sqrt {\frac {1}{8}}}\!\,}
5
14
{\displaystyle {\sqrt {\frac {5}{14}}}\!\,}
1
2
{\displaystyle {\sqrt {\frac {1}{2}}}\!\,}
3 / 2 , -1 / 2
15
56
{\displaystyle {\sqrt {\frac {15}{56}}}\!\,}
49
120
{\displaystyle {\sqrt {\frac {49}{120}}}\!\,}
1
42
{\displaystyle {\sqrt {\frac {1}{42}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
1 / 2 , 1 / 2
15
28
{\displaystyle {\sqrt {\frac {15}{28}}}\!\,}
−
1
60
{\displaystyle -{\sqrt {\frac {1}{60}}}\!\,}
−
25
84
{\displaystyle -{\sqrt {\frac {25}{84}}}\!\,}
3
20
{\displaystyle {\sqrt {\frac {3}{20}}}\!\,}
-1 / 2 , 3 / 2
5
28
{\displaystyle {\sqrt {\frac {5}{28}}}\!\,}
−
9
20
{\displaystyle -{\sqrt {\frac {9}{20}}}\!\,}
9
28
{\displaystyle {\sqrt {\frac {9}{28}}}\!\,}
−
1
20
{\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}
m =0
j
m 1 , m 2
4
3
2
1
3 / 2 , -3 / 2
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
1 / 2 , -1 / 2
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
−
1
14
{\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
-1 / 2 , 1 / 2
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
−
1
14
{\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}
3
10
{\displaystyle {\sqrt {\frac {3}{10}}}\!\,}
-3 / 2 , 3 / 2
1
14
{\displaystyle {\sqrt {\frac {1}{14}}}\!\,}
−
3
10
{\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
−
1
5
{\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}
j 1 =5 / 2 , j 2 =2
m =9 / 2
j
m 1 , m 2
9 / 2
5 / 2 , 2
1
{\displaystyle 1\!\,}
m =7 / 2
j
m 1 , m 2
9 / 2
7 / 2
5 / 2 , 1
2
3
{\displaystyle {\frac {2}{3}}\!\,}
5
9
{\displaystyle {\sqrt {\frac {5}{9}}}\!\,}
3 / 2 , 2
5
9
{\displaystyle {\sqrt {\frac {5}{9}}}\!\,}
−
2
3
{\displaystyle -{\frac {2}{3}}\!\,}
m =5 / 2
j
m 1 , m 2
9 / 2
7 / 2
5 / 2
5 / 2 , 0
1
6
{\displaystyle {\sqrt {\frac {1}{6}}}\!\,}
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
5
14
{\displaystyle {\sqrt {\frac {5}{14}}}\!\,}
3 / 2 , 1
5
9
{\displaystyle {\sqrt {\frac {5}{9}}}\!\,}
1
63
{\displaystyle {\sqrt {\frac {1}{63}}}\!\,}
−
3
7
{\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}
1 / 2 , 2
5
18
{\displaystyle {\sqrt {\frac {5}{18}}}\!\,}
−
32
63
{\displaystyle -{\sqrt {\frac {32}{63}}}\!\,}
3
14
{\displaystyle {\sqrt {\frac {3}{14}}}\!\,}
m =3 / 2
j
m 1 , m 2
9 / 2
7 / 2
5 / 2
3 / 2
5 / 2 , -1
1
21
{\displaystyle {\sqrt {\frac {1}{21}}}\!\,}
5
21
{\displaystyle {\sqrt {\frac {5}{21}}}\!\,}
3
7
{\displaystyle {\sqrt {\frac {3}{7}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
3 / 2 , 0
5
14
{\displaystyle {\sqrt {\frac {5}{14}}}\!\,}
2
7
{\displaystyle {\sqrt {\frac {2}{7}}}\!\,}
−
1
70
{\displaystyle -{\sqrt {\frac {1}{70}}}\!\,}
−
12
35
{\displaystyle -{\sqrt {\frac {12}{35}}}\!\,}
1 / 2 , 1
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
−
2
21
{\displaystyle -{\sqrt {\frac {2}{21}}}\!\,}
−
6
35
{\displaystyle -{\sqrt {\frac {6}{35}}}\!\,}
9
35
{\displaystyle {\sqrt {\frac {9}{35}}}\!\,}
-1 / 2 , 2
5
42
{\displaystyle {\sqrt {\frac {5}{42}}}\!\,}
−
8
21
{\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}
27
70
{\displaystyle {\sqrt {\frac {27}{70}}}\!\,}
−
4
35
{\displaystyle -{\sqrt {\frac {4}{35}}}\!\,}
m =1 / 2
j
m 1 , m 2
9 / 2
7 / 2
5 / 2
3 / 2
1 / 2
5 / 2 , -2
1
126
{\displaystyle {\sqrt {\frac {1}{126}}}\!\,}
4
63
{\displaystyle {\sqrt {\frac {4}{63}}}\!\,}
3
14
{\displaystyle {\sqrt {\frac {3}{14}}}\!\,}
8
21
{\displaystyle {\sqrt {\frac {8}{21}}}\!\,}
1
3
{\displaystyle {\sqrt {\frac {1}{3}}}\!\,}
3 / 2 , -1
10
63
{\displaystyle {\sqrt {\frac {10}{63}}}\!\,}
121
315
{\displaystyle {\sqrt {\frac {121}{315}}}\!\,}
6
35
{\displaystyle {\sqrt {\frac {6}{35}}}\!\,}
−
2
105
{\displaystyle -{\sqrt {\frac {2}{105}}}\!\,}
−
4
15
{\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}
1 / 2 , 0
10
21
{\displaystyle {\sqrt {\frac {10}{21}}}\!\,}
4
105
{\displaystyle {\sqrt {\frac {4}{105}}}\!\,}
−
8
35
{\displaystyle -{\sqrt {\frac {8}{35}}}\!\,}
−
2
35
{\displaystyle -{\sqrt {\frac {2}{35}}}\!\,}
1
5
{\displaystyle {\sqrt {\frac {1}{5}}}\!\,}
-1 / 2 , 1
20
63
{\displaystyle {\sqrt {\frac {20}{63}}}\!\,}
−
14
45
{\displaystyle -{\sqrt {\frac {14}{45}}}\!\,}
0
{\displaystyle 0\!\,}
5
21
{\displaystyle {\sqrt {\frac {5}{21}}}\!\,}
−
2
15
{\displaystyle -{\sqrt {\frac {2}{15}}}\!\,}
-3 / 2 , 2
5
126
{\displaystyle {\sqrt {\frac {5}{126}}}\!\,}
−
64
315
{\displaystyle -{\sqrt {\frac {64}{315}}}\!\,}
27
70
{\displaystyle {\sqrt {\frac {27}{70}}}\!\,}
−
32
105
{\displaystyle -{\sqrt {\frac {32}{105}}}\!\,}
1
15
{\displaystyle {\sqrt {\frac {1}{15}}}\!\,}
SU(N) Clebsch–Gordan coefficients
Algorithms to produce Clebsch–Gordan coefficients for higher values of
j
1
{\displaystyle j_{1}}
and
j
2
{\displaystyle j_{2}}
, or for the su(N) algebra instead of su(2), are known.[6]
A web interface for tabulating SU(N) Clebsch-Gordan coefficients is readily available.
References
^ Baird, C.E.; L. C. Biedenharn (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn " . J. Math. Phys . 5 : 1723–1730. Bibcode :1964JMP.....5.1723B . doi :10.1063/1.1704095 . Retrieved 2007-12-20 .
^ Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF) . Phys. Rev. D . 66 : 010001. Bibcode :2002PhRvD..66a0001H . doi :10.1103/PhysRevD.66.010001 . Retrieved 2007-12-20 .
^ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text) . Retrieved 2012-10-15 .
^ (2.41), p. 172, Quantum Mechanics: Foundations and Applications , Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, ISBN 0-387-95330-2.
^ Weisbluth, Michael (1978). Atoms and molecules . ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5 . Table 1.4 resumes the most common.
^ Alex, A.; M. Kalus; A. Huckleberry; J. von Delft (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients" . J. Math. Phys . 82 : 023507. arXiv :1009.0437 . Bibcode :2011JMP....52b3507A . doi :10.1063/1.3521562 . Retrieved 2011-04-13 .
External links