# Table of Clebsch–Gordan coefficients

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 ${\displaystyle j_{1}}$, ${\displaystyle j_{2}}$, ${\displaystyle j}$ 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

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

Explicitly:

{\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 j1 < j2 are omitted. They may be calculated using the simple relations

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

and

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

## A complete list [5]

### j2=0

When j2 = 0, the Clebsch–Gordan coefficients are given by ${\displaystyle \delta _{j,j_{1}}\delta _{m,m_{1}}}$ .

### j1=1/2, j2=1/2

m=1 j
m1m2
1 ${\displaystyle 1\!\,}$
m=-1 j
m1m2
1 ${\displaystyle 1\!\,}$
m=0 j
m1m2
1 0 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

### j1=1, j2=1/2

m=3/2 j
m1m2
3/2 ${\displaystyle 1\!\,}$
m=1/2 j
m1m2
3/2 1/2 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$

### j1=1, j2=1

m=2 j
m1m2
2 ${\displaystyle 1\!\,}$
m=1 j
m1m2
2 1 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$
m=0 j
m1m2
2 1 0 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$

### j1=3/2, j2=1/2

m=2 j
m1m2
2 ${\displaystyle 1\!\,}$
m=1 j
m1m2
2 1 ${\displaystyle {\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{4}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{4}}}\!\,}$ ${\displaystyle -{\frac {1}{2}}\!\,}$
m=0 j
m1m2
2 1 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

### j1=3/2, j2=1

m=5/2 j
m1m2
5/2 ${\displaystyle 1\!\,}$
m=3/2 j
m1m2
5/2 3/2 ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$
m=1/2 j
m1m2
5/2 3/2 1/2 ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {8}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$

### j1=3/2, j2=3/2

m=3 j
m1m2
3 ${\displaystyle 1\!\,}$
m=2 j
m1m2
3 2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$
m=1 j
m1m2
3 2 1 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$
m=0 j
m1m2
3 2 1 0 ${\displaystyle {\sqrt {\frac {1}{20}}}\!\,}$ ${\displaystyle {\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$ ${\displaystyle {\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$ ${\displaystyle {\frac {1}{2}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$ ${\displaystyle -{\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$ ${\displaystyle -{\frac {1}{2}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$ ${\displaystyle {\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{20}}}\!\,}$ ${\displaystyle -{\frac {1}{2}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{20}}}\!\,}$ ${\displaystyle -{\frac {1}{2}}\!\,}$

### j1=2, j2=1/2

m=5/2 j
m1m2
5/2 ${\displaystyle 1\!\,}$
m=3/2 j
m1m2
5/2 3/2 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$
m=1/2 j
m1m2
5/2 3/2 ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$

### j1=2, j2=1

m=3 j
m1m2
3 ${\displaystyle 1\!\,}$
m=2 j
m1m2
3 2 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$
m=1 j
m1m2
3 2 1 ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {8}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$
m=0 j
m1m2
3 2 1 ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{5}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$

### j1=2, j2=3/2

m=7/2 j
m1m2
7/2 ${\displaystyle 1\!\,}$
m=5/2 j
m1m2
7/2 5/2 ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$
m=3/2 j
m1m2
7/2 5/2 3/2 ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {16}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$
m=1/2 j
m1m2
7/2 5/2 3/2 1/2 ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {6}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {12}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {18}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {27}{70}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$

### j1=2, j2=2

m=4 j
m1m2
4 ${\displaystyle 1\!\,}$
m=3 j
m1m2
4 3 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$
m=2 j
m1m2
4 3 2 ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$
m=1 j
m1m2
4 3 2 1 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$
m=0 j
m1m2
4 3 2 1 0 ${\displaystyle {\sqrt {\frac {1}{70}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {8}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {18}{35}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {8}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{10}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{70}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$

### j1=5/2, j2=1/2

m=3 j
m1m2
3 ${\displaystyle 1\!\,}$
m=2 j
m1m2
3 2 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{6}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{6}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{6}}}\!\,}$
m=1 j
m1m2
3 2 ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{3}}}\!\,}$
m=0 j
m1m2
3 2 ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$

### j1=5/2, j2=1

m=7/2 j
m1m2
7/2 ${\displaystyle 1\!\,}$
m=5/2 j
m1m2
7/2 5/2 ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{7}}}\!\,}$
m=3/2 j
m1m2
7/2 5/2 3/2 ${\displaystyle {\sqrt {\frac {1}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {16}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$
m=1/2 j
m1m2
7/2 5/2 3/2 ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {16}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {18}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$

### j1=5/2, j2=3/2

m=4 j
m1m2
4 ${\displaystyle 1\!\,}$
m=3 j
m1m2
4 3 ${\displaystyle {\sqrt {\frac {3}{8}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{8}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{8}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{8}}}\!\,}$
m=2 j
m1m2
4 3 2 ${\displaystyle {\sqrt {\frac {3}{28}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{12}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {15}{28}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{12}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{7}}}\!\,}$
m=1 j
m1m2
4 3 2 1 ${\displaystyle {\sqrt {\frac {1}{56}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{8}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{2}}}\!\,}$ ${\displaystyle {\sqrt {\frac {15}{56}}}\!\,}$ ${\displaystyle {\sqrt {\frac {49}{120}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{42}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {15}{28}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{60}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {25}{84}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{20}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{28}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {9}{20}}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{28}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{20}}}\!\,}$
m=0 j
m1m2
4 3 2 1 ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{14}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{10}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{5}}}\!\,}$

### j1=5/2, j2=2

m=9/2 j
m1m2
9/2 ${\displaystyle 1\!\,}$
m=7/2 j
m1m2
9/2 7/2 ${\displaystyle {\frac {2}{3}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$ ${\displaystyle -{\frac {2}{3}}\!\,}$
m=5/2 j
m1m2
9/2 7/2 5/2 ${\displaystyle {\sqrt {\frac {1}{6}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{9}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{63}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{18}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {32}{63}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$
m=3/2 j
m1m2
9/2 7/2 5/2 3/2 ${\displaystyle {\sqrt {\frac {1}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {2}{7}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {1}{70}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {12}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{21}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {6}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {9}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{42}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {8}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {27}{70}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {4}{35}}}\!\,}$
m=1/2 j
m1m2
9/2 7/2 5/2 3/2 1/2 ${\displaystyle {\sqrt {\frac {1}{126}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{63}}}\!\,}$ ${\displaystyle {\sqrt {\frac {3}{14}}}\!\,}$ ${\displaystyle {\sqrt {\frac {8}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{3}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{63}}}\!\,}$ ${\displaystyle {\sqrt {\frac {121}{315}}}\!\,}$ ${\displaystyle {\sqrt {\frac {6}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{105}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {4}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {10}{21}}}\!\,}$ ${\displaystyle {\sqrt {\frac {4}{105}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {8}{35}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{35}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{5}}}\!\,}$ ${\displaystyle {\sqrt {\frac {20}{63}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {14}{45}}}\!\,}$ ${\displaystyle 0\!\,}$ ${\displaystyle {\sqrt {\frac {5}{21}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {2}{15}}}\!\,}$ ${\displaystyle {\sqrt {\frac {5}{126}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {64}{315}}}\!\,}$ ${\displaystyle {\sqrt {\frac {27}{70}}}\!\,}$ ${\displaystyle -{\sqrt {\frac {32}{105}}}\!\,}$ ${\displaystyle {\sqrt {\frac {1}{15}}}\!\,}$

## SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of ${\displaystyle j_{1}}$ and ${\displaystyle j_{2}}$, 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

1. ^ 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.
2. ^ 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.
3. ^ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). Retrieved 2012-10-15.
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.
5. ^ Weisbluth, Michael (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. Table 1.4 resumes the most common.
6. ^ 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:. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13.