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 $\delta _{j,j_{1}}\delta _{m,m_{1}}$ .

$j1 = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/2⁠, j2 = ⁠1/2⁠$

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

[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.

[1]

[2]

[3]

[4]

[5]

[6]

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