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 $j_{1}$ , $j_{2}$ , $j$ 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\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}\mid j_{1},j_{2};J,M\rangle

Explicitly:

{\begin{aligned}&\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle \\[6pt]={}&\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 {}\\[6pt]&{\sqrt {(J+M)!(J-M)!(j_{1}-m_{1})!(j_{1}+m_{1})!(j_{2}-m_{2})!(j_{2}+m_{2})!}}\ \times {}\\[6pt]&\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

\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{1},j_{2};-m_{1},-m_{2}\mid j_{1},j_{2};J,-M\rangle .

and

\langle j_{1},j_{2};m_{1},m_{2}\mid j_{1},j_{2};J,M\rangle =(-1)^{J-j_{1}-j_{2}}\langle j_{2},j_{1};m_{2},m_{1}\mid j_{2},j_{1};J,M\rangle .

Specific values

The Clebsch–Gordan coefficients for j values less than or equal to 5/2 are given below.[5]

$j 2 = 0$

When $j 2 = 0$ , the Clebsch–Gordan coefficients are given by $\delta _{j,j_{1}}\delta _{m,m_{1}}$ .

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$m = 1$
$j$ $m 1, m 2$	1
1/2, 1/2	$1$

$m = -1$
$j$ $m 1, m 2$	1
−1/2, −1/2	$1$

$m = 0$
$j$ $m 1, m 2$	1	0
1/2, −1/2	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−1/2, 1/2	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

$j 1 = 1, j 2 = 1 / 2$

$m = 3 / 2$
$j$ $m 1, m 2$	3/2
1, 1/2	$1$

$m = 1 / 2$
$j$ $m 1, m 2$	3/2	1/2
1, −1/2	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {2}{3}}}$
0, 1/2	${\sqrt {\frac {2}{3}}}$	$-{\sqrt {\frac {1}{3}}}$

$j 1 = 1, j 2 = 1$

$m = 2$
$j$ $m 1, m 2$	2
1, 1	$1$

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

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

$j 1 = 3 / 2, j 2 = 1 / 2$

$m = 2$
$j$ $m 1, m 2$	2
3/2, 1/2	$1$

$m = 1$
$j$ $m 1, m 2$	2	1
3/2, −1/2	${\frac {1}{2}}$	${\sqrt {\frac {3}{4}}}$
1/2, 1/2	${\sqrt {\frac {3}{4}}}$	$-{\frac {1}{2}}$

$m = 0$
$j$ $m 1, m 2$	2	1
1/2, −1/2	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−1/2, 1/2	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

$j 1 = 3 / 2, j 2 = 1$

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
3/2, 1	$1$

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

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

$j 1 = 3 / 2, j 2 = 3 / 2$

$m = 3$
$j$ $m 1, m 2$	3
3/2, 3/2	$1$

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

$m = 1$
$j$ $m 1, m 2$	3	2	1
3/2, −1/2	${\sqrt {\frac {1}{5}}}$	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {3}{10}}}$
1/2, 1/2	${\sqrt {\frac {3}{5}}}$	$0$	$-{\sqrt {\frac {2}{5}}}$
−1/2, 3/2	${\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {3}{10}}}$

$m = 0$
$j$ $m 1, m 2$	3	2	1	0
3/2, −3/2	${\sqrt {\frac {1}{20}}}$	${\frac {1}{2}}$	${\sqrt {\frac {9}{20}}}$	${\frac {1}{2}}$
1/2, −1/2	${\sqrt {\frac {9}{20}}}$	${\frac {1}{2}}$	$-{\sqrt {\frac {1}{20}}}$	$-{\frac {1}{2}}$
−1/2, 1/2	${\sqrt {\frac {9}{20}}}$	$-{\frac {1}{2}}$	$-{\sqrt {\frac {1}{20}}}$	${\frac {1}{2}}$
−3/2, 3/2	${\sqrt {\frac {1}{20}}}$	$-{\frac {1}{2}}$	${\sqrt {\frac {9}{20}}}$	$-{\frac {1}{2}}$

$j 1 = 2, j 2 = 1 / 2$

$m = 5 / 2$
$j$ $m 1, m 2$	5/2
2, 1/2	$1$

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

$m = 1 / 2$
$j$ $m 1, m 2$	5/2	3/2
1, −1/2	${\sqrt {\frac {2}{5}}}$	${\sqrt {\frac {3}{5}}}$
0, 1/2	${\sqrt {\frac {3}{5}}}$	$-{\sqrt {\frac {2}{5}}}$

$j 1 = 2, j 2 = 1$

$m = 3$
$j$ $m 1, m 2$	3
2, 1	$1$

$m = 2$
$j$ $m 1, m 2$	3	2
2, 0	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {2}{3}}}$
1, 1	${\sqrt {\frac {2}{3}}}$	$-{\sqrt {\frac {1}{3}}}$

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

$m = 0$
$j$ $m 1, m 2$	3	2	1
1, −1	${\sqrt {\frac {1}{5}}}$	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {3}{10}}}$
0, 0	${\sqrt {\frac {3}{5}}}$	$0$	$-{\sqrt {\frac {2}{5}}}$
−1, 1	${\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {3}{10}}}$

$j 1 = 2, j 2 = 3 / 2$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
2, 3/2	$1$

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

$m = 3 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
2, −1/2	${\sqrt {\frac {1}{7}}}$	${\sqrt {\frac {16}{35}}}$	${\sqrt {\frac {2}{5}}}$
1, 1/2	${\sqrt {\frac {4}{7}}}$	${\sqrt {\frac {1}{35}}}$	$-{\sqrt {\frac {2}{5}}}$
0, 3/2	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {18}{35}}}$	${\sqrt {\frac {1}{5}}}$

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

$j 1 = 2, j 2 = 2$

$m = 4$
$j$ $m 1, m 2$	4
2, 2	$1$

$m = 3$
$j$ $m 1, m 2$	4	3
2, 1	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
1, 2	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

$m = 2$
$j$ $m 1, m 2$	4	3	2
2, 0	${\sqrt {\frac {3}{14}}}$	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {2}{7}}}$
1, 1	${\sqrt {\frac {4}{7}}}$	$0$	$-{\sqrt {\frac {3}{7}}}$
0, 2	${\sqrt {\frac {3}{14}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {2}{7}}}$

$m = 1$
$j$ $m 1, m 2$	4	3	2	1
2, −1	${\sqrt {\frac {1}{14}}}$	${\sqrt {\frac {3}{10}}}$	${\sqrt {\frac {3}{7}}}$	${\sqrt {\frac {1}{5}}}$
1, 0	${\sqrt {\frac {3}{7}}}$	${\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{14}}}$	$-{\sqrt {\frac {3}{10}}}$
0, 1	${\sqrt {\frac {3}{7}}}$	$-{\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{14}}}$	${\sqrt {\frac {3}{10}}}$
−1, 2	${\sqrt {\frac {1}{14}}}$	$-{\sqrt {\frac {3}{10}}}$	${\sqrt {\frac {3}{7}}}$	$-{\sqrt {\frac {1}{5}}}$

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

$j 1 = 5 / 2, j 2 = 1 / 2$

$m = 3$
$j$ $m 1, m 2$	3
5/2, 1/2	$1$

$m = 2$
$j$ $m 1, m 2$	3	2
5/2, −1/2	${\sqrt {\frac {1}{6}}}$	${\sqrt {\frac {5}{6}}}$
3/2, 1/2	${\sqrt {\frac {5}{6}}}$	$-{\sqrt {\frac {1}{6}}}$

$m = 1$
$j$ $m 1, m 2$	3	2
3/2, −1/2	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {2}{3}}}$
1/2, 1/2	${\sqrt {\frac {2}{3}}}$	$-{\sqrt {\frac {1}{3}}}$

$m = 0$
$j$ $m 1, m 2$	3	2
1/2, −1/2	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−1/2, 1/2	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

$j 1 = 5 / 2, j 2 = 1$

$m = 7 / 2$
$j$ $m 1, m 2$	7/2
5/2, 1	$1$

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

$m = 3 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
5/2, −1	${\sqrt {\frac {1}{21}}}$	${\sqrt {\frac {2}{7}}}$	${\sqrt {\frac {2}{3}}}$
3/2, 0	${\sqrt {\frac {10}{21}}}$	${\sqrt {\frac {9}{35}}}$	$-{\sqrt {\frac {4}{15}}}$
1/2, 1	${\sqrt {\frac {10}{21}}}$	$-{\sqrt {\frac {16}{35}}}$	${\sqrt {\frac {1}{15}}}$

$m = 1 / 2$
$j$ $m 1, m 2$	7/2	5/2	3/2
3/2, −1	${\sqrt {\frac {1}{7}}}$	${\sqrt {\frac {16}{35}}}$	${\sqrt {\frac {2}{5}}}$
1/2, 0	${\sqrt {\frac {4}{7}}}$	${\sqrt {\frac {1}{35}}}$	$-{\sqrt {\frac {2}{5}}}$
−1/2, 1	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {18}{35}}}$	${\sqrt {\frac {1}{5}}}$

$j 1 = 5 / 2, j 2 = 3 / 2$

$m = 4$
$j$ $m 1, m 2$	4
5/2, 3/2	$1$

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

$m = 2$
$j$ $m 1, m 2$	4	3	2
5/2, −1/2	${\sqrt {\frac {3}{28}}}$	${\sqrt {\frac {5}{12}}}$	${\sqrt {\frac {10}{21}}}$
3/2, 1/2	${\sqrt {\frac {15}{28}}}$	${\sqrt {\frac {1}{12}}}$	$-{\sqrt {\frac {8}{21}}}$
1/2, 3/2	${\sqrt {\frac {5}{14}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{7}}}$

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

$m = 0$
$j$ $m 1, m 2$	4	3	2	1
3/2, −3/2	${\sqrt {\frac {1}{14}}}$	${\sqrt {\frac {3}{10}}}$	${\sqrt {\frac {3}{7}}}$	${\sqrt {\frac {1}{5}}}$
1/2, −1/2	${\sqrt {\frac {3}{7}}}$	${\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{14}}}$	$-{\sqrt {\frac {3}{10}}}$
−1/2, 1/2	${\sqrt {\frac {3}{7}}}$	$-{\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{14}}}$	${\sqrt {\frac {3}{10}}}$
−3/2, 3/2	${\sqrt {\frac {1}{14}}}$	$-{\sqrt {\frac {3}{10}}}$	${\sqrt {\frac {3}{7}}}$	$-{\sqrt {\frac {1}{5}}}$

$j 1 = 5 / 2, j 2 = 2$

$m = 9 / 2$
$j$ $m 1, m 2$	9/2
5/2, 2	$1$

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

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

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

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

$j 1 = 5 / 2, j 2 = 5 / 2$

$m = 5$
$j$ $m 1, m 2$	5
5/2, 5/2	$1$

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

$m = 3$
$j$ $m 1, m 2$	5	4	3
5/2, 1/2	${\sqrt {\frac {2}{9}}}$	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {5}{18}}}$
3/2, 3/2	${\sqrt {\frac {5}{9}}}$	$0$	$-{\frac {2}{3}}$
1/2, 5/2	${\sqrt {\frac {2}{9}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {5}{18}}}$

$m = 2$
$j$ $m 1, m 2$	5	4	3	2
5/2, −1/2	${\sqrt {\frac {1}{12}}}$	${\sqrt {\frac {9}{28}}}$	${\sqrt {\frac {5}{12}}}$	${\sqrt {\frac {5}{28}}}$
3/2, 1/2	${\sqrt {\frac {5}{12}}}$	${\sqrt {\frac {5}{28}}}$	$-{\sqrt {\frac {1}{12}}}$	$-{\sqrt {\frac {9}{28}}}$
1/2, 3/2	${\sqrt {\frac {5}{12}}}$	$-{\sqrt {\frac {5}{28}}}$	$-{\sqrt {\frac {1}{12}}}$	${\sqrt {\frac {9}{28}}}$
−1/2, 5/2	${\sqrt {\frac {1}{12}}}$	$-{\sqrt {\frac {9}{28}}}$	${\sqrt {\frac {5}{12}}}$	$-{\sqrt {\frac {5}{28}}}$

$m = 1$
$j$ $m 1, m 2$	5	4	3	2	1
5/2, −3/2	${\sqrt {\frac {1}{42}}}$	${\sqrt {\frac {1}{7}}}$	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {5}{14}}}$	${\sqrt {\frac {1}{7}}}$
3/2, −1/2	${\sqrt {\frac {5}{21}}}$	${\sqrt {\frac {5}{14}}}$	${\sqrt {\frac {1}{30}}}$	$-{\sqrt {\frac {1}{7}}}$	$-{\sqrt {\frac {8}{35}}}$
1/2, 1/2	${\sqrt {\frac {10}{21}}}$	$0$	$-{\sqrt {\frac {4}{15}}}$	$0$	${\sqrt {\frac {9}{35}}}$
−1/2, 3/2	${\sqrt {\frac {5}{21}}}$	$-{\sqrt {\frac {5}{14}}}$	${\sqrt {\frac {1}{30}}}$	${\sqrt {\frac {1}{7}}}$	$-{\sqrt {\frac {8}{35}}}$
−3/2, 5/2	${\sqrt {\frac {1}{42}}}$	$-{\sqrt {\frac {1}{7}}}$	${\sqrt {\frac {1}{3}}}$	$-{\sqrt {\frac {5}{14}}}$	${\sqrt {\frac {1}{7}}}$

$m = 0$
$j$ $m 1, m 2$	5	4	3	2	1	0
5/2, −5/2	${\sqrt {\frac {1}{252}}}$	${\sqrt {\frac {1}{28}}}$	${\sqrt {\frac {5}{36}}}$	${\sqrt {\frac {25}{84}}}$	${\sqrt {\frac {5}{14}}}$	${\sqrt {\frac {1}{6}}}$
3/2, −3/2	${\sqrt {\frac {25}{252}}}$	${\sqrt {\frac {9}{28}}}$	${\sqrt {\frac {49}{180}}}$	${\sqrt {\frac {1}{84}}}$	$-{\sqrt {\frac {9}{70}}}$	$-{\sqrt {\frac {1}{6}}}$
1/2, −1/2	${\sqrt {\frac {25}{63}}}$	${\sqrt {\frac {1}{7}}}$	$-{\sqrt {\frac {4}{45}}}$	$-{\sqrt {\frac {4}{21}}}$	${\sqrt {\frac {1}{70}}}$	${\sqrt {\frac {1}{6}}}$
−1/2, 1/2	${\sqrt {\frac {25}{63}}}$	$-{\sqrt {\frac {1}{7}}}$	$-{\sqrt {\frac {4}{45}}}$	${\sqrt {\frac {4}{21}}}$	${\sqrt {\frac {1}{70}}}$	$-{\sqrt {\frac {1}{6}}}$
−3/2, 3/2	${\sqrt {\frac {25}{252}}}$	$-{\sqrt {\frac {9}{28}}}$	${\sqrt {\frac {49}{180}}}$	$-{\sqrt {\frac {1}{84}}}$	$-{\sqrt {\frac {9}{70}}}$	${\sqrt {\frac {1}{6}}}$
−5/2, 5/2	${\sqrt {\frac {1}{252}}}$	$-{\sqrt {\frac {1}{28}}}$	${\sqrt {\frac {5}{36}}}$	$-{\sqrt {\frac {25}{84}}}$	${\sqrt {\frac {5}{14}}}$	$-{\sqrt {\frac {1}{6}}}$

SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of $j_{1}$ and $j_{2}$ , 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 SU_n". J. Math. Phys. 5 (12): 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095.
Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF). Phys. Rev. D. 66 (1): 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.
Weissbluth, Mitchel (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.

External links

Online, Java-based Clebsch–Gordan Coefficient Calculator by Paul Stevenson
Other formulae for Clebsch–Gordan coefficients.
Web interface for tabulating SU(N) Clebsch–Gordan coefficients

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[1] Baird, C.E.; L. C. Biedenharn (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_n". J. Math. Phys. 5 (12): 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095.

[2] Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF). Phys. Rev. D. 66 (1): 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] Weissbluth, Mitchel (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:1009.0437. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562.

Table of Clebsch–Gordan coefficients

Formulation

Specific values

j2 = 0

j1 = 1, j2 = 1/2

j1 = 1, j2 = 1

j1 = 3/2, j2 = 1/2

j1 = 3/2, j2 = 1

j1 = 3/2, j2 = 3/2

j1 = 2, j2 = 1/2

j1 = 2, j2 = 1

j1 = 2, j2 = 3/2

j1 = 2, j2 = 2

j1 = 5/2, j2 = 1/2

j1 = 5/2, j2 = 1

j1 = 5/2, j2 = 3/2

j1 = 5/2, j2 = 2

j1 = 5/2, j2 = 5/2