De Wikipedia, la enciclopedia libre
Para ver su formulación y definición formal vea coeficientes Clebsch—Gordan .Esta es una tabla de coeficientes de Clebsch-Gordan usada para sumar momentos angulares en mecánica cuántica . El signo global de los coeficientes para cada conjunto de
j
1
{\displaystyle j_{1}}
j
2
{\displaystyle j_{2}}
j
{\displaystyle j}
[ 1] Review of Particle Properties
[ 2] Particle Data Group
y en tablas en línea.[ 3]
Los estados de momento angular total pueden ser expandidos usando la relación de completitud en las bases de momentos angulares parciales como
|
(
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
m
1
j
2
m
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
m
1
j
2
m
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}m_{1}j_{2}m_{2}|JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle }
Los coeficientes
⟨
j
1
m
1
j
2
m
2
|
J
M
⟩
{\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle }
coeficientes de Clebsch–Gordan . Estos se pueden escribir explícitamente como
⟨
j
1
m
1
j
2
m
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
)
!
×
{\displaystyle \langle j_{1}m_{1}j_{2}m_{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 }
(
j
+
m
)
!
(
j
−
m
)
!
(
j
1
−
m
1
)
!
(
j
1
+
m
1
)
!
(
j
2
−
m
2
)
!
(
j
2
+
m
2
)
!
×
{\displaystyle {\sqrt {(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times }
∑
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 \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)!}}.}
k enteros para los para los que los argumentos de todos los factoriales involucrados sean no negativos.[ 4]
m
<
0
{\displaystyle m<0}
⟨
j
1
,
m
1
,
j
2
,
m
2
|
j
,
m
⟩
=
(
−
1
)
j
−
j
1
−
j
2
⟨
j
1
,
−
m
1
,
j
2
,
−
m
2
|
j
,
−
m
⟩
{\displaystyle \langle j_{1},m_{1},j_{2},m_{2}|j,m\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{1},-m_{1},j_{2},-m_{2}|j,-m\rangle }
m=1
j=
1 , m2 =
1
1/2, 1/2
1
{\displaystyle 1\!\,}
m=0
j=
1 , m2 =
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}}}\!\,}
m=3/2
j=
1 , m2 =
3/2
1, 1/2
1
{\displaystyle 1\!\,}
m=1/2
j=
1 , m2 =
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}}}\!\,}
m=2
j=
1 , m2 =
2
1, 1
1
{\displaystyle 1\!\,}
m=1
j=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=2
j=
1 , m2 =
2
3/2, 1/2
1
{\displaystyle 1\!\,}
m=1
j=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=5/2
j=
1 , m2 =
5/2
3/2, 1
1
{\displaystyle 1\!\,}
m=3/2
j=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=3
j=
1 , m2 =
3
3/2, 3/2
1
{\displaystyle 1\!\,}
m=2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}\!\,}
m=5/2
j=
1 , m2 =
5/2
2, 1/2
1
{\displaystyle 1\!\,}
m=3/2
j=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=3
j=
1 , m2 =
3
2, 1
1
{\displaystyle 1\!\,}
m=2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=7/2
j=
1 , m2 =
7/2
2, 3/2
1
{\displaystyle 1\!\,}
m=5/2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=4
j=
1 , m2 =
4
2, 2
1
{\displaystyle 1\!\,}
m=3
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=3
j=
1 , m2 =
3
5/2, 1/2
1
{\displaystyle 1\!\,}
m=2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=7/2
j=
1 , m2 =
7/2
5/2, 1
1
{\displaystyle 1\!\,}
m=5/2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=4
j=
1 , m2 =
4
5/2, 3/2
1
{\displaystyle 1\!\,}
m=3
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
m=9/2
j=
1 , m2 =
9/2
5/2, 2
1
{\displaystyle 1\!\,}
m=7/2
j=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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=
1 , m2 =
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}}}\!\,}
↑ Baird, C.E.; L. C. Biedenharn (octubre de 1964). «On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn » . J. Math. Phys. 5 : 1723-1730. doi :10.1063/1.1704095 . Consultado el 20 de diciembre de 2007 . ↑ Hagiwara, K.; et al. (julio de 2002). «Review of Particle Properties» (PDF) . Phys. Rev. D 66 : 010001. doi :10.1103/PhysRevD.66.010001 . Consultado el 20 de diciembre de 2007 . ↑ Mathar, Richard J. (14 de agosto de 2006). «SO(3) Clebsch Gordan coefficients» (txt) . Consultado el 20 de diciembre de 2007 . ↑ (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 .
