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]

[1]

[2]

[3]

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

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

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

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

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

Table of Clebsch–Gordan coefficients

Formulation

Specific values

$j 2 = 0$

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

$j 1 = 1, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 3 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = 2, j 2 = 1$

$j 1 = 2, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = 2, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 1 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 1$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 3 / 2 ⁠$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = 2$

$j 1 = ⁠ 5 / 2 ⁠, j 2 = ⁠ 5 / 2 ⁠$

SU(N) Clebsch–Gordan coefficients

References

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

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

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

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

$j$ $m 1, m 2$	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	${\sqrt {\frac {1}{20}}}$	$-{\frac {1}{2}}$	${\sqrt {\frac {9}{20}}}$	$-{\frac {1}{2}}$

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

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

$j$ $m 1, m 2$	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	${\sqrt {\frac {2}{5}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{10}}}$

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

$j$ $m 1, m 2$	⁠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	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {18}{35}}}$	${\sqrt {\frac {1}{5}}}$

$j$ $m 1, m 2$	⁠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	${\sqrt {\frac {4}{35}}}$	$-{\sqrt {\frac {27}{70}}}$	${\sqrt {\frac {2}{5}}}$	$-{\sqrt {\frac {1}{10}}}$

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

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

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

$j$ $m 1, m 2$	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	${\sqrt {\frac {1}{70}}}$	$-{\sqrt {\frac {1}{10}}}$	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {2}{5}}}$	${\sqrt {\frac {1}{5}}}$

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

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

$j$ $m 1, m 2$	⁠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	${\sqrt {\frac {10}{21}}}$	$-{\sqrt {\frac {16}{35}}}$	${\sqrt {\frac {1}{15}}}$

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

$j$ $m 1, m 2$	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	${\sqrt {\frac {5}{14}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{7}}}$

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