Table of Clebsch–Gordan coefficients

Used for adding angular momentum values in quantum mechanics

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. Tables with the same sign convention may be found in the Particle Data Group's Review of Particle Properties and in online tables.

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 \\={}&\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.

For brevity, solutions with m < 0 and j₁ < j₂ 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.

j₂ = 0

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

j₁ = ⁠1/2⁠, j₂ = ⁠1/2⁠

m = 1
jm₁, m₂	1
⁠1/2⁠, ⁠1/2⁠	$1$

m = −1
jm₁, m₂	1
−⁠1/2⁠, −⁠1/2⁠	$1$

m = 0
jm₁, m₂	1	0
⁠1/2⁠, −⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−⁠1/2⁠, ⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

j₁ = 1, j₂ = 1/2

m = ⁠3/2⁠
jm₁, m₂	⁠3/2⁠
1, ⁠1/2⁠	$1$

m = ⁠1/2⁠
jm₁, m₂	⁠3/2⁠	⁠1/2⁠
1, −⁠1/2⁠	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {2}{3}}}$
0, ⁠1/2⁠	${\sqrt {\frac {2}{3}}}$	$-{\sqrt {\frac {1}{3}}}$

j₁ = 1, j₂ = 1

m = 2
jm₁, m₂	2
1, 1	$1$

m = 1
jm₁, m₂	2	1
1, 0	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
0, 1	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

m = 0
jm₁, m₂	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	${\sqrt {\frac {1}{6}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{3}}}$

j₁ = ⁠3/2⁠, j₂ = ⁠1/2⁠

m = 2
jm₁, m₂	2
⁠3/2⁠, ⁠1/2⁠	$1$

m = 1
jm₁, m₂	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
jm₁, m₂	2	1
⁠1/2⁠, −⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−⁠1/2⁠, ⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

j₁ = ⁠3/2⁠, j₂ = 1

m = ⁠5/2⁠
jm₁, m₂	⁠5/2⁠
⁠3/2⁠, 1	$1$

m = ⁠3/2⁠
jm₁, m₂	⁠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
jm₁, m₂	⁠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	${\sqrt {\frac {3}{10}}}$	$-{\sqrt {\frac {8}{15}}}$	${\sqrt {\frac {1}{6}}}$

j₁ = ⁠3/2⁠, j₂ = ⁠3/2⁠

m = 3
jm₁, m₂	3
⁠3/2⁠, ⁠3/2⁠	$1$

m = 2
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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	${\sqrt {\frac {1}{20}}}$	$-{\frac {1}{2}}$	${\sqrt {\frac {9}{20}}}$	$-{\frac {1}{2}}$

j₁ = 2, j₂ = ⁠1/2⁠

m = ⁠5/2⁠
jm₁, m₂	⁠5/2⁠
2, ⁠1/2⁠	$1$

m = ⁠3/2⁠
jm₁, m₂	⁠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
jm₁, m₂	⁠5/2⁠	⁠3/2⁠
1, −⁠1/2⁠	${\sqrt {\frac {2}{5}}}$	${\sqrt {\frac {3}{5}}}$
0, ⁠1/2⁠	${\sqrt {\frac {3}{5}}}$	$-{\sqrt {\frac {2}{5}}}$

j₁ = 2, j₂ = 1

m = 3
jm₁, m₂	3
2, 1	$1$

m = 2
jm₁, m₂	3	2
2, 0	${\sqrt {\frac {1}{3}}}$	${\sqrt {\frac {2}{3}}}$
1, 1	${\sqrt {\frac {2}{3}}}$	$-{\sqrt {\frac {1}{3}}}$

m = 1
jm₁, m₂	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
jm₁, m₂	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	${\sqrt {\frac {1}{5}}}$	$-{\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {3}{10}}}$

j₁ = 2, j₂ = ⁠3/2⁠

m = ⁠7/2⁠
jm₁, m₂	⁠7/2⁠
2, ⁠3/2⁠	$1$

m = ⁠5/2⁠
jm₁, m₂	⁠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
jm₁, m₂	⁠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
jm₁, m₂	⁠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	${\sqrt {\frac {4}{35}}}$	$-{\sqrt {\frac {27}{70}}}$	${\sqrt {\frac {2}{5}}}$	$-{\sqrt {\frac {1}{10}}}$

j₁ = 2, j₂ = 2

m = 4
jm₁, m₂	4
2, 2	$1$

m = 3
jm₁, m₂	4	3
2, 1	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
1, 2	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

m = 2
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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	${\sqrt {\frac {1}{70}}}$	$-{\sqrt {\frac {1}{10}}}$	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {2}{5}}}$	${\sqrt {\frac {1}{5}}}$

j₁ = ⁠5/2⁠, j₂ = ⁠1/2⁠

m = 3
jm₁, m₂	3
⁠5/2⁠, ⁠1/2⁠	$1$

m = 2
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	3	2
⁠1/2⁠, −⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	${\sqrt {\frac {1}{2}}}$
−⁠1/2⁠, ⁠1/2⁠	${\sqrt {\frac {1}{2}}}$	$-{\sqrt {\frac {1}{2}}}$

j₁ = ⁠5/2⁠, j₂ = 1

m = ⁠7/2⁠
jm₁, m₂	⁠7/2⁠
⁠5/2⁠, 1	$1$

m = ⁠5/2⁠
jm₁, m₂	⁠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
jm₁, m₂	⁠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
jm₁, m₂	⁠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	${\sqrt {\frac {2}{7}}}$	$-{\sqrt {\frac {18}{35}}}$	${\sqrt {\frac {1}{5}}}$

j₁ = ⁠5/2⁠, j₂ = ⁠3/2⁠

m = 4
jm₁, m₂	4
⁠5/2⁠, ⁠3/2⁠	$1$

m = 3
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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	${\sqrt {\frac {1}{14}}}$	$-{\sqrt {\frac {3}{10}}}$	${\sqrt {\frac {3}{7}}}$	$-{\sqrt {\frac {1}{5}}}$

j₁ = ⁠5/2⁠, j₂ = 2

m = ⁠9/2⁠
jm₁, m₂	⁠9/2⁠
⁠5/2⁠, 2	$1$

m = ⁠7/2⁠
jm₁, m₂	⁠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
jm₁, m₂	⁠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
jm₁, m₂	⁠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
jm₁, m₂	⁠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	${\sqrt {\frac {5}{126}}}$	$-{\sqrt {\frac {64}{315}}}$	${\sqrt {\frac {27}{70}}}$	$-{\sqrt {\frac {32}{105}}}$	${\sqrt {\frac {1}{15}}}$

j₁ = ⁠5/2⁠, j₂ = ⁠5/2⁠

m = 5
jm₁, m₂	5
⁠5/2⁠, ⁠5/2⁠	$1$

m = 4
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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
jm₁, m₂	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. 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

Categories:

Formulation

Specific values

j2 = 0

j1 = ⁠1/2⁠, j2 = ⁠1/2⁠

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