Anexo:Tabla de coeficientes de Clebsch-Gordan

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 ${\displaystyle j_{1}}$, ${\displaystyle j_{2}}$ y ${\displaystyle j}$ constantes es en cierto grado arbitrario y ha sido fijado de acuerdo con la convención de Condon-Shortley y Wigner como viene dada en Baird and Biedenharn .^[1]​ Tablas con la misma convención de signos pueden ser encontradas en Review of Particle Properties ^[2]​ del Particle Data Group y en tablas en línea.^[3]​

Formulación

Los estados de momento angular total pueden ser expandidos usando la relación de completitud en las bases de momentos angulares parciales como

${\displaystyle |(j_{1}j_{2})JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle =\sum _{m_{1}=-j_{1}}^{j_{1}}\sum _{m_{2}=-j_{2}}^{j_{2}}|j_{1}m_{1}\rangle |j_{2}m_{2}\rangle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle }$ 

Los coeficientes ${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|JM\rangle }$  de la expansión son los coeficientes de Clebsch–Gordan. Estos se pueden escribir explícitamente como

${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|jm\rangle =\delta _{m,m_{1}+m_{2}}{\sqrt {\frac {(2j+1)(j+j_{1}-j_{2})!(j-j_{1}+j_{2})!(j_{1}+j_{2}-j)!}{(j_{1}+j_{2}+j+1)!}}}\ \times }$ 

${\displaystyle {\sqrt {(j+m)!(j-m)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times }$ 

${\displaystyle \sum _{k}{\frac {(-1)^{k}}{k!(j_{1}+j_{2}-j-k)!(j_{1}-m_{1}-k)!(j_{2}+m_{2}-k)!(j-j_{2}+m_{1}+k)!(j-j_{1}-m_{2}+k)!}}.}$ 

Donde la suma se hace sobre todos los k enteros para los para los que los argumentos de todos los factoriales involucrados sean no negativos.^[4]​ Por brevedad, las soluciones con ${\displaystyle m<0}$  son omitidas, pero pueden ser calculadas usando la siguiente relación

${\displaystyle \langle j_{1},m_{1},j_{2},m_{2}|j,m\rangle =(-1)^{j-j_{1}-j_{2}}\langle j_{1},-m_{1},j_{2},-m_{2}|j,-m\rangle }$  .

j₁=1/2, j₂=1/2

m=1	j=
m1, m2=	1 1/2, 1/2 ${\displaystyle 1\!\,}$

m=0	j=
m1, m2=	1 0 1/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  -1/2, 1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

j₁=1, j₂=1/2

m=3/2	j=
m1, m2=	3/2 1, 1/2 ${\displaystyle 1\!\,}$

m=1/2	j=
m1, m2=	3/2 1/2 1, -1/2 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  0, 1/2 ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$

j₁=1, j₂=1

m=2	j=
m1, m2=	2 1, 1 ${\displaystyle 1\!\,}$

m=1	j=
m1, m2=	2 1 1, 0 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  0, 1 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

m=0

j=

m1, m2=

2 1 0 1, -1 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  0, 0 ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$  -1, 1 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$

j₁=3/2, j₂=1/2

m=2	j=
m1, m2=	2 3/2, 1/2 ${\displaystyle 1\!\,}$

m=1	j=
m1, m2=	2 1 3/2, -1/2 ${\displaystyle {\frac {1}{2}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{4}}}\!\,}$  1/2, 1/2 ${\displaystyle {\sqrt {\frac {3}{4}}}\!\,}$  ${\displaystyle -{\frac {1}{2}}\!\,}$

m=0	j=
m1, m2=	2 1 1/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  -1/2, 1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

j₁=3/2, j₂=1

m=5/2	j=
m1, m2=	5/2 3/2, 1 ${\displaystyle 1\!\,}$

m=3/2	j=
m1, m2=	5/2 3/2 3/2, 0 ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  1/2, 1 ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$

m=1/2

j=

m1, m2=

5/2 3/2 1/2 3/2, -1 ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  1/2, 0 ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$  -1/2, 1 ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {8}{15}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$

j₁=3/2, j₂=3/2

m=3	j=
m1, m2=	3 3/2, 3/2 ${\displaystyle 1\!\,}$

m=2	j=
m1, m2=	3 2 3/2, 1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  1/2, 3/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

m=1

j=

m1, m2=

3 2 1 3/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  1/2, 1/2 ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  -1/2, 3/2 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$

m=0

j=

m1, m2=

3 2 1 0 3/2, -3/2 ${\displaystyle {\sqrt {\frac {1}{20}}}\!\,}$  ${\displaystyle {\frac {1}{2}}\!\,}$  ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$  ${\displaystyle {\frac {1}{2}}\!\,}$  1/2, -1/2 ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$  ${\displaystyle {\frac {1}{2}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$  ${\displaystyle -{\frac {1}{2}}\!\,}$  -1/2, 1/2 ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$  ${\displaystyle -{\frac {1}{2}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$  ${\displaystyle {\frac {1}{2}}\!\,}$  -3/2, 3/2 ${\displaystyle {\sqrt {\frac {1}{20}}}\!\,}$  ${\displaystyle -{\frac {1}{2}}\!\,}$  ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$  ${\displaystyle -{\frac {1}{2}}\!\,}$

j₁=2, j₂=1/2

m=5/2	j=
m1, m2=	5/2 2, 1/2 ${\displaystyle 1\!\,}$

m=3/2	j=
m1, m2=	5/2 3/2 2, -1/2 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {4}{5}}}\!\,}$  1, 1/2 ${\displaystyle {\sqrt {\frac {4}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$

m=1/2	j=
m1, m2=	5/2 3/2 1, -1/2 ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  0, 1/2 ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$

j₁=2, j₂=1

m=3	j=
m1, m2=	3 2, 1 ${\displaystyle 1\!\,}$

m=2	j=
m1, m2=	3 2 2, 0 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  1, 1 ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$

m=1

j=

m1, m2=

3 2 1 2, -1 ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  1, 0 ${\displaystyle {\sqrt {\frac {8}{15}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  0, 1 ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$

m=0

j=

m1, m2=

3 2 1 1, -1 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  0, 0 ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  -1, 1 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$

j₁=2, j₂=3/2

m=7/2	j=
m1, m2=	7/2 2, 3/2 ${\displaystyle 1\!\,}$

m=5/2	j=
m1, m2=	7/2 5/2 2, 1/2 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$  1, 3/2 ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$

m=3/2

j=

m1, m2=

7/2 5/2 3/2 2, -1/2 ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {16}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  1, 1/2 ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  0, 3/2 ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$

m=1/2

j=

m1, m2=

7/2 5/2 3/2 1/2 2, -3/2 ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {6}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  1, -1/2 ${\displaystyle {\sqrt {\frac {12}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  0, 1/2 ${\displaystyle {\sqrt {\frac {18}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  -1, 3/2 ${\displaystyle {\sqrt {\frac {4}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {27}{70}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$

j₁=2, j₂=2

m=4	j=
m1, m2=	4 2, 2 ${\displaystyle 1\!\,}$

m=3	j=
m1, m2=	4 3 2, 1 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  1, 2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

m=2

j=

m1, m2=

4 3 2 2, 0 ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  1, 1 ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$  0, 2 ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$

m=1

j=

m1, m2=

4 3 2 1 2, -1 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  1, 0 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  0, 1 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  -1, 2 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$

m=0

j=

m1, m2=

4 3 2 1 0 2, -2 ${\displaystyle {\sqrt {\frac {1}{70}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  1, -1 ${\displaystyle {\sqrt {\frac {8}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$  0, 0 ${\displaystyle {\sqrt {\frac {18}{35}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  -1, 1 ${\displaystyle {\sqrt {\frac {8}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$  -2, 2 ${\displaystyle {\sqrt {\frac {1}{70}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$

j₁=5/2, j₂=1/2

m=3	j=
m1, m2=	3 5/2, 1/2 ${\displaystyle 1\!\,}$

m=2	j=
m1, m2=	3 2 5/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{6}}}\!\,}$  3/2, 1/2 ${\displaystyle {\sqrt {\frac {5}{6}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{6}}}\!\,}$

m=1	j=
m1, m2=	3 2 3/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  1/2, 1/2 ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$

m=0	j=
m1, m2=	3 2 1/2, -1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  -1/2, 1/2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

j₁=5/2, j₂=1

m=7/2	j=
m1, m2=	7/2 5/2, 1 ${\displaystyle 1\!\,}$

m=5/2	j=
m1, m2=	7/2 5/2 5/2, 0 ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{7}}}\!\,}$  3/2, 1 ${\displaystyle {\sqrt {\frac {5}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}$

m=3/2

j=

m1, m2=

7/2 5/2 3/2 5/2, -1 ${\displaystyle {\sqrt {\frac {1}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$  3/2, 0 ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {9}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}$  1/2, 1 ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {16}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$

m=1/2

j=

m1, m2=

7/2 5/2 3/2 3/2, -1 ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {16}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$  1/2, 0 ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$  -1/2, 1 ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$

j₁=5/2, j₂=3/2

m=4	j=
m1, m2=	4 5/2, 3/2 ${\displaystyle 1\!\,}$

m=3	j=
m1, m2=	4 3 5/2, 1/2 ${\displaystyle {\sqrt {\frac {3}{8}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{8}}}\!\,}$  3/2, 3/2 ${\displaystyle {\sqrt {\frac {5}{8}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{8}}}\!\,}$

m=2

j=

m1, m2=

4 3 2 5/2, -1/2 ${\displaystyle {\sqrt {\frac {3}{28}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{12}}}\!\,}$  ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  3/2, 1/2 ${\displaystyle {\sqrt {\frac {15}{28}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{12}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}$  1/2, 3/2 ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$

m=1

j=

m1, m2=

4 3 2 1 5/2, -3/2 ${\displaystyle {\sqrt {\frac {1}{56}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{8}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$  3/2, -1/2 ${\displaystyle {\sqrt {\frac {15}{56}}}\!\,}$  ${\displaystyle {\sqrt {\frac {49}{120}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{42}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  1/2, 1/2 ${\displaystyle {\sqrt {\frac {15}{28}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{60}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {25}{84}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{20}}}\!\,}$  -1/2, 3/2 ${\displaystyle {\sqrt {\frac {5}{28}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {9}{20}}}\!\,}$  ${\displaystyle {\sqrt {\frac {9}{28}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$

m=0

j=

m1, m2=

4 3 2 1 3/2, -3/2 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  1/2, -1/2 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  -1/2, 1/2 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$  -3/2, 3/2 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$

j₁=5/2, j₂=2

m=9/2	j=
m1, m2=	9/2 5/2, 2 ${\displaystyle 1\!\,}$

m=7/2	j=
m1, m2=	9/2 7/2 5/2, 1 ${\displaystyle {\frac {2}{3}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$  3/2, 2 ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$  ${\displaystyle -{\frac {2}{3}}\!\,}$

m=5/2

j=

m1, m2=

9/2 7/2 5/2 5/2, 0 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$  ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$  3/2, 1 ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{63}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$  1/2, 2 ${\displaystyle {\sqrt {\frac {5}{18}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {32}{63}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$

m=3/2

j=

m1, m2=

9/2 7/2 5/2 3/2 5/2, -1 ${\displaystyle {\sqrt {\frac {1}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {5}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  3/2, 0 ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {1}{70}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {12}{35}}}\!\,}$  1/2, 1 ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{21}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {6}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {9}{35}}}\!\,}$  -1/2, 2 ${\displaystyle {\sqrt {\frac {5}{42}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {27}{70}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {4}{35}}}\!\,}$

m=1/2

j=

m1, m2=

9/2 7/2 5/2 3/2 1/2 5/2, -2 ${\displaystyle {\sqrt {\frac {1}{126}}}\!\,}$  ${\displaystyle {\sqrt {\frac {4}{63}}}\!\,}$  ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$  ${\displaystyle {\sqrt {\frac {8}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$  3/2, -1 ${\displaystyle {\sqrt {\frac {10}{63}}}\!\,}$  ${\displaystyle {\sqrt {\frac {121}{315}}}\!\,}$  ${\displaystyle {\sqrt {\frac {6}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{105}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}$  1/2, 0 ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$  ${\displaystyle {\sqrt {\frac {4}{105}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {8}{35}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{35}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$  -1/2, 1 ${\displaystyle {\sqrt {\frac {20}{63}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {14}{45}}}\!\,}$  ${\displaystyle 0\!\,}$  ${\displaystyle {\sqrt {\frac {5}{21}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {2}{15}}}\!\,}$  -3/2, 2 ${\displaystyle {\sqrt {\frac {5}{126}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {64}{315}}}\!\,}$  ${\displaystyle {\sqrt {\frac {27}{70}}}\!\,}$  ${\displaystyle -{\sqrt {\frac {32}{105}}}\!\,}$  ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$

Referencias

1. Baird, C.E.; L. C. Biedenharn (octubre de 1964). «On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_n». J. Math. Phys. 5: 1723-1730. doi:10.1063/1.1704095. Consultado el 20 de diciembre de 2007.  La referencia utiliza el parámetro obsoleto |coautores= (ayuda)
2. 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.  La referencia utiliza el parámetro obsoleto |coautores= (ayuda)
3. Mathar, Richard J. (14 de agosto de 2006). «SO(3) Clebsch Gordan coefficients» (txt). Consultado el 20 de diciembre de 2007. 
4. (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.

Enlaces externos

• Calculadora de coeficientes de Clebsch-Gordan online basada en Java, de Paul Stevenson.
• Otras fórmulas de coeficientes de Clebsch-Gordan.