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

| j 1 , j 2 ; J , M ⟩ = ∑ m 1 = − j 1 j 1 ∑ m 2 = − j 2 j 2 | j 1 , m 1 ; j 2 , m 2 ⟩ ⟨ j 1 , j 2 ; m 1 , m 2 ∣ j 1 , j 2 ; J , M ⟩ {\displaystyle |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:

⟨ j 1 , j 2 ; m 1 , m 2 ∣ j 1 , j 2 ; J , M ⟩ = δ M , m 1 + m 2 ( 2 J + 1 ) ( J + j 1 − j 2 ) ! ( J − j 1 + j 2 ) ! ( j 1 + j 2 − J ) ! ( j 1 + j 2 + J + 1 ) !   × ( J + M ) ! ( J − M ) ! ( j 1 − m 1 ) ! ( j 1 + m 1 ) ! ( j 2 − m 2 ) ! ( j 2 + m 2 ) !   × ∑ k ( − 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 ) ! . {\displaystyle {\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 j1 < j2 are omitted. They may be calculated using the simple relations

⟨ j 1 , j 2 ; m 1 , m 2 ∣ j 1 , j 2 ; J , M ⟩ = ( − 1 ) J − j 1 − j 2 ⟨ j 1 , j 2 ; − m 1 , − m 2 ∣ j 1 , j 2 ; J , − M ⟩ . {\displaystyle \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

⟨ j 1 , j 2 ; m 1 , m 2 ∣ j 1 , j 2 ; J , M ⟩ = ( − 1 ) J − j 1 − j 2 ⟨ j 2 , j 1 ; m 2 , m 1 ∣ j 2 , j 1 ; J , M ⟩ . {\displaystyle \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]

 j2 = 0

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

 j1 = ⁠1/2⁠,  j2 = ⁠1/2⁠

m = 1
j
m1, m2
 1
⁠1/2⁠, ⁠1/2⁠ 1 {\displaystyle 1}
m = −1
j
m1, m2
 1
−⁠1/2⁠, −⁠1/2⁠ 1 {\displaystyle 1}
m = 0
j
m1, m2
 1 0
⁠1/2⁠, −⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
−⁠1/2⁠, ⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}

 j1 = 1,  j2 = 1/2

m = ⁠3/2⁠
j
m1, m2
 ⁠3/2⁠
1, ⁠1/2⁠ 1 {\displaystyle 1}
m = ⁠1/2⁠
j
m1, m2
 ⁠3/2⁠ ⁠1/2⁠
1, −⁠1/2⁠ 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}}
0, ⁠1/2⁠ 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}} − 1 3 {\displaystyle -{\sqrt {\frac {1}{3}}}}

 j1 = 1,  j2 = 1

m = 2
j
m1, m2
 2
1, 1 1 {\displaystyle 1}
m = 1
j
m1, m2
 2 1
1, 0 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
0, 1 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}
m = 0
j
m1, m2
 2 1 0
1, −1 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}}
0, 0 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}} 0 {\displaystyle 0} − 1 3 {\displaystyle -{\sqrt {\frac {1}{3}}}}
−1, 1 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}}

 j1 = ⁠3/2⁠,  j2 = ⁠1/2⁠

m = 2
j
m1, m2
 2
⁠3/2⁠, ⁠1/2⁠ 1 {\displaystyle 1}
m = 1
j
m1, m2
 2 1
⁠3/2⁠, −⁠1/2⁠ 1 2 {\displaystyle {\frac {1}{2}}} 3 4 {\displaystyle {\sqrt {\frac {3}{4}}}}
⁠1/2⁠, ⁠1/2⁠ 3 4 {\displaystyle {\sqrt {\frac {3}{4}}}} − 1 2 {\displaystyle -{\frac {1}{2}}}
m = 0
j
m1, m2
 2 1
⁠1/2⁠, −⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
−⁠1/2⁠, ⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}

 j1 = ⁠3/2⁠,  j2 = 1

m = ⁠5/2⁠
j
m1, m2
 ⁠5/2⁠
⁠3/2⁠, 1 1 {\displaystyle 1}
m = ⁠3/2⁠
j
m1, m2
 ⁠5/2⁠ ⁠3/2⁠
⁠3/2⁠, 0 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}}
⁠1/2⁠, 1 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}
m = ⁠1/2⁠
j
m1, m2
 ⁠5/2⁠ ⁠3/2⁠ ⁠1/2⁠
⁠3/2⁠, −1 1 10 {\displaystyle {\sqrt {\frac {1}{10}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
⁠1/2⁠, 0 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}} 1 15 {\displaystyle {\sqrt {\frac {1}{15}}}} − 1 3 {\displaystyle -{\sqrt {\frac {1}{3}}}}
−⁠1/2⁠, 1 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}} − 8 15 {\displaystyle -{\sqrt {\frac {8}{15}}}} 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}}

 j1 = ⁠3/2⁠,  j2 = ⁠3/2⁠

m = 3
j
m1, m2
 3
⁠3/2⁠, ⁠3/2⁠ 1 {\displaystyle 1}
m = 2
j
m1, m2
 3 2
⁠3/2⁠, ⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
⁠1/2⁠, ⁠3/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}
m = 1
j
m1, m2
 3 2 1
⁠3/2⁠, −⁠1/2⁠ 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}
⁠1/2⁠, ⁠1/2⁠ 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}} 0 {\displaystyle 0} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}
−⁠1/2⁠, ⁠3/2⁠ 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}
m = 0
j
m1, m2
 3 2 1 0
⁠3/2⁠, −⁠3/2⁠ 1 20 {\displaystyle {\sqrt {\frac {1}{20}}}} 1 2 {\displaystyle {\frac {1}{2}}} 9 20 {\displaystyle {\sqrt {\frac {9}{20}}}} 1 2 {\displaystyle {\frac {1}{2}}}
⁠1/2⁠, −⁠1/2⁠ 9 20 {\displaystyle {\sqrt {\frac {9}{20}}}} 1 2 {\displaystyle {\frac {1}{2}}} − 1 20 {\displaystyle -{\sqrt {\frac {1}{20}}}} − 1 2 {\displaystyle -{\frac {1}{2}}}
−⁠1/2⁠, ⁠1/2⁠ 9 20 {\displaystyle {\sqrt {\frac {9}{20}}}} − 1 2 {\displaystyle -{\frac {1}{2}}} − 1 20 {\displaystyle -{\sqrt {\frac {1}{20}}}} 1 2 {\displaystyle {\frac {1}{2}}}
−⁠3/2⁠, ⁠3/2⁠ 1 20 {\displaystyle {\sqrt {\frac {1}{20}}}} − 1 2 {\displaystyle -{\frac {1}{2}}} 9 20 {\displaystyle {\sqrt {\frac {9}{20}}}} − 1 2 {\displaystyle -{\frac {1}{2}}}

 j1 = 2,  j2 = ⁠1/2⁠

m = ⁠5/2⁠
j
m1, m2
 ⁠5/2⁠
2, ⁠1/2⁠ 1 {\displaystyle 1}
m = ⁠3/2⁠
j
m1, m2
 ⁠5/2⁠ ⁠3/2⁠
2, −⁠1/2⁠ 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} 4 5 {\displaystyle {\sqrt {\frac {4}{5}}}}
1, ⁠1/2⁠ 4 5 {\displaystyle {\sqrt {\frac {4}{5}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}}
m = ⁠1/2⁠
j
m1, m2
 ⁠5/2⁠ ⁠3/2⁠
1, −⁠1/2⁠ 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}}
0, ⁠1/2⁠ 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}

 j1 = 2,  j2 = 1

m = 3
j
m1, m2
 3
2, 1 1 {\displaystyle 1}
m = 2
j
m1, m2
 3 2
2, 0 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}}
1, 1 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}} − 1 3 {\displaystyle -{\sqrt {\frac {1}{3}}}}
m = 1
j
m1, m2
 3 2 1
2, −1 1 15 {\displaystyle {\sqrt {\frac {1}{15}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}}
1, 0 8 15 {\displaystyle {\sqrt {\frac {8}{15}}}} 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}}
0, 1 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 1 10 {\displaystyle {\sqrt {\frac {1}{10}}}}
m = 0
j
m1, m2
 3 2 1
1, −1 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}
0, 0 3 5 {\displaystyle {\sqrt {\frac {3}{5}}}} 0 {\displaystyle 0} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}
−1, 1 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}

 j1 = 2,  j2 = ⁠3/2⁠

m = ⁠7/2⁠
j
m1, m2
 ⁠7/2⁠
2, ⁠3/2⁠ 1 {\displaystyle 1}
m = ⁠5/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠
2, ⁠1/2⁠ 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 4 7 {\displaystyle {\sqrt {\frac {4}{7}}}}
1, ⁠3/2⁠ 4 7 {\displaystyle {\sqrt {\frac {4}{7}}}} − 3 7 {\displaystyle -{\sqrt {\frac {3}{7}}}}
m = ⁠3/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠
2, −⁠1/2⁠ 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}} 16 35 {\displaystyle {\sqrt {\frac {16}{35}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}}
1, ⁠1/2⁠ 4 7 {\displaystyle {\sqrt {\frac {4}{7}}}} 1 35 {\displaystyle {\sqrt {\frac {1}{35}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}
0, ⁠3/2⁠ 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} − 18 35 {\displaystyle -{\sqrt {\frac {18}{35}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
m = ⁠1/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠ ⁠1/2⁠
2, −⁠3/2⁠ 1 35 {\displaystyle {\sqrt {\frac {1}{35}}}} 6 35 {\displaystyle {\sqrt {\frac {6}{35}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}}
1, −⁠1/2⁠ 12 35 {\displaystyle {\sqrt {\frac {12}{35}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 0 {\displaystyle 0} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}}
0, ⁠1/2⁠ 18 35 {\displaystyle {\sqrt {\frac {18}{35}}}} − 3 35 {\displaystyle -{\sqrt {\frac {3}{35}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
−1, ⁠3/2⁠ 4 35 {\displaystyle {\sqrt {\frac {4}{35}}}} − 27 70 {\displaystyle -{\sqrt {\frac {27}{70}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} − 1 10 {\displaystyle -{\sqrt {\frac {1}{10}}}}

 j1 = 2,  j2 = 2

m = 4
j
m1, m2
 4
2, 2 1 {\displaystyle 1}
m = 3
j
m1, m2
 4 3
2, 1 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
1, 2 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}
m = 2
j
m1, m2
 4 3 2
2, 0 3 14 {\displaystyle {\sqrt {\frac {3}{14}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}}
1, 1 4 7 {\displaystyle {\sqrt {\frac {4}{7}}}} 0 {\displaystyle 0} − 3 7 {\displaystyle -{\sqrt {\frac {3}{7}}}}
0, 2 3 14 {\displaystyle {\sqrt {\frac {3}{14}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}}
m = 1
j
m1, m2
 4 3 2 1
2, −1 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}} 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
1, 0 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} − 1 14 {\displaystyle -{\sqrt {\frac {1}{14}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}}
0, 1 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}} − 1 14 {\displaystyle -{\sqrt {\frac {1}{14}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}
−1, 2 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}} 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}}
m = 0
j
m1, m2
 4 3 2 1 0
2, −2 1 70 {\displaystyle {\sqrt {\frac {1}{70}}}} 1 10 {\displaystyle {\sqrt {\frac {1}{10}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
1, −1 8 35 {\displaystyle {\sqrt {\frac {8}{35}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}} 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} − 1 10 {\displaystyle -{\sqrt {\frac {1}{10}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}}
0, 0 18 35 {\displaystyle {\sqrt {\frac {18}{35}}}} 0 {\displaystyle 0} − 2 7 {\displaystyle -{\sqrt {\frac {2}{7}}}} 0 {\displaystyle 0} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
−1, 1 8 35 {\displaystyle {\sqrt {\frac {8}{35}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}} 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} 1 10 {\displaystyle {\sqrt {\frac {1}{10}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}}
−2, 2 1 70 {\displaystyle {\sqrt {\frac {1}{70}}}} − 1 10 {\displaystyle -{\sqrt {\frac {1}{10}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}

 j1 = ⁠5/2⁠,  j2 = ⁠1/2⁠

m = 3
j
m1, m2
 3
⁠5/2⁠, ⁠1/2⁠ 1 {\displaystyle 1}
m = 2
j
m1, m2
 3 2
⁠5/2⁠, −⁠1/2⁠ 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}} 5 6 {\displaystyle {\sqrt {\frac {5}{6}}}}
⁠3/2⁠, ⁠1/2⁠ 5 6 {\displaystyle {\sqrt {\frac {5}{6}}}} − 1 6 {\displaystyle -{\sqrt {\frac {1}{6}}}}
m = 1
j
m1, m2
 3 2
⁠3/2⁠, −⁠1/2⁠ 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}}
⁠1/2⁠, ⁠1/2⁠ 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}} − 1 3 {\displaystyle -{\sqrt {\frac {1}{3}}}}
m = 0
j
m1, m2
 3 2
⁠1/2⁠, −⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
−⁠1/2⁠, ⁠1/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}

 j1 = ⁠5/2⁠,  j2 = 1

m = ⁠7/2⁠
j
m1, m2
 ⁠7/2⁠
⁠5/2⁠, 1 1 {\displaystyle 1}
m = ⁠5/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠
⁠5/2⁠, 0 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} 5 7 {\displaystyle {\sqrt {\frac {5}{7}}}}
⁠3/2⁠, 1 5 7 {\displaystyle {\sqrt {\frac {5}{7}}}} − 2 7 {\displaystyle -{\sqrt {\frac {2}{7}}}}
m = ⁠3/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠
⁠5/2⁠, −1 1 21 {\displaystyle {\sqrt {\frac {1}{21}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} 2 3 {\displaystyle {\sqrt {\frac {2}{3}}}}
⁠3/2⁠, 0 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} 9 35 {\displaystyle {\sqrt {\frac {9}{35}}}} − 4 15 {\displaystyle -{\sqrt {\frac {4}{15}}}}
⁠1/2⁠, 1 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} − 16 35 {\displaystyle -{\sqrt {\frac {16}{35}}}} 1 15 {\displaystyle {\sqrt {\frac {1}{15}}}}
m = ⁠1/2⁠
j
m1, m2
 ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠
⁠3/2⁠, −1 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}} 16 35 {\displaystyle {\sqrt {\frac {16}{35}}}} 2 5 {\displaystyle {\sqrt {\frac {2}{5}}}}
⁠1/2⁠, 0 4 7 {\displaystyle {\sqrt {\frac {4}{7}}}} 1 35 {\displaystyle {\sqrt {\frac {1}{35}}}} − 2 5 {\displaystyle -{\sqrt {\frac {2}{5}}}}
−⁠1/2⁠, 1 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} − 18 35 {\displaystyle -{\sqrt {\frac {18}{35}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}

 j1 = ⁠5/2⁠,  j2 = ⁠3/2⁠

m = 4
j
m1, m2
 4
⁠5/2⁠, ⁠3/2⁠ 1 {\displaystyle 1}
m = 3
j
m1, m2
 4 3
⁠5/2⁠, ⁠1/2⁠ 3 8 {\displaystyle {\sqrt {\frac {3}{8}}}} 5 8 {\displaystyle {\sqrt {\frac {5}{8}}}}
⁠3/2⁠, ⁠3/2⁠ 5 8 {\displaystyle {\sqrt {\frac {5}{8}}}} − 3 8 {\displaystyle -{\sqrt {\frac {3}{8}}}}
m = 2
j
m1, m2
 4 3 2
⁠5/2⁠, −⁠1/2⁠ 3 28 {\displaystyle {\sqrt {\frac {3}{28}}}} 5 12 {\displaystyle {\sqrt {\frac {5}{12}}}} 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}}
⁠3/2⁠, ⁠1/2⁠ 15 28 {\displaystyle {\sqrt {\frac {15}{28}}}} 1 12 {\displaystyle {\sqrt {\frac {1}{12}}}} − 8 21 {\displaystyle -{\sqrt {\frac {8}{21}}}}
⁠1/2⁠, ⁠3/2⁠ 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}}
m = 1
j
m1, m2
 4 3 2 1
⁠5/2⁠, −⁠3/2⁠ 1 56 {\displaystyle {\sqrt {\frac {1}{56}}}} 1 8 {\displaystyle {\sqrt {\frac {1}{8}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
⁠3/2⁠, −⁠1/2⁠ 15 56 {\displaystyle {\sqrt {\frac {15}{56}}}} 49 120 {\displaystyle {\sqrt {\frac {49}{120}}}} 1 42 {\displaystyle {\sqrt {\frac {1}{42}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}}
⁠1/2⁠, ⁠1/2⁠ 15 28 {\displaystyle {\sqrt {\frac {15}{28}}}} − 1 60 {\displaystyle -{\sqrt {\frac {1}{60}}}} − 25 84 {\displaystyle -{\sqrt {\frac {25}{84}}}} 3 20 {\displaystyle {\sqrt {\frac {3}{20}}}}
−⁠1/2⁠, ⁠3/2⁠ 5 28 {\displaystyle {\sqrt {\frac {5}{28}}}} − 9 20 {\displaystyle -{\sqrt {\frac {9}{20}}}} 9 28 {\displaystyle {\sqrt {\frac {9}{28}}}} − 1 20 {\displaystyle -{\sqrt {\frac {1}{20}}}}
m = 0
j
m1, m2
 4 3 2 1
⁠3/2⁠, −⁠3/2⁠ 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}} 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
⁠1/2⁠, −⁠1/2⁠ 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}} − 1 14 {\displaystyle -{\sqrt {\frac {1}{14}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}}
−⁠1/2⁠, ⁠1/2⁠ 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}} − 1 14 {\displaystyle -{\sqrt {\frac {1}{14}}}} 3 10 {\displaystyle {\sqrt {\frac {3}{10}}}}
−⁠3/2⁠, ⁠3/2⁠ 1 14 {\displaystyle {\sqrt {\frac {1}{14}}}} − 3 10 {\displaystyle -{\sqrt {\frac {3}{10}}}} 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} − 1 5 {\displaystyle -{\sqrt {\frac {1}{5}}}}

 j1 = ⁠5/2⁠,  j2 = 2

m = ⁠9/2⁠
j
m1, m2
 ⁠9/2⁠
⁠5/2⁠, 2 1 {\displaystyle 1}
m = ⁠7/2⁠
j
m1, m2
 ⁠9/2⁠ ⁠7/2⁠
⁠5/2⁠, 1 2 3 {\displaystyle {\frac {2}{3}}} 5 9 {\displaystyle {\sqrt {\frac {5}{9}}}}
⁠3/2⁠, 2 5 9 {\displaystyle {\sqrt {\frac {5}{9}}}} − 2 3 {\displaystyle -{\frac {2}{3}}}
m = ⁠5/2⁠
j
m1, m2
 ⁠9/2⁠ ⁠7/2⁠ ⁠5/2⁠
⁠5/2⁠, 0 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}} 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}}
⁠3/2⁠, 1 5 9 {\displaystyle {\sqrt {\frac {5}{9}}}} 1 63 {\displaystyle {\sqrt {\frac {1}{63}}}} − 3 7 {\displaystyle -{\sqrt {\frac {3}{7}}}}
⁠1/2⁠, 2 5 18 {\displaystyle {\sqrt {\frac {5}{18}}}} − 32 63 {\displaystyle -{\sqrt {\frac {32}{63}}}} 3 14 {\displaystyle {\sqrt {\frac {3}{14}}}}
m = ⁠3/2⁠
j
m1, m2
 ⁠9/2⁠ ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠
⁠5/2⁠, −1 1 21 {\displaystyle {\sqrt {\frac {1}{21}}}} 5 21 {\displaystyle {\sqrt {\frac {5}{21}}}} 3 7 {\displaystyle {\sqrt {\frac {3}{7}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}}
⁠3/2⁠, 0 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 2 7 {\displaystyle {\sqrt {\frac {2}{7}}}} − 1 70 {\displaystyle -{\sqrt {\frac {1}{70}}}} − 12 35 {\displaystyle -{\sqrt {\frac {12}{35}}}}
⁠1/2⁠, 1 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} − 2 21 {\displaystyle -{\sqrt {\frac {2}{21}}}} − 6 35 {\displaystyle -{\sqrt {\frac {6}{35}}}} 9 35 {\displaystyle {\sqrt {\frac {9}{35}}}}
−⁠1/2⁠, 2 5 42 {\displaystyle {\sqrt {\frac {5}{42}}}} − 8 21 {\displaystyle -{\sqrt {\frac {8}{21}}}} 27 70 {\displaystyle {\sqrt {\frac {27}{70}}}} − 4 35 {\displaystyle -{\sqrt {\frac {4}{35}}}}
m = ⁠1/2⁠
j
m1, m2
 ⁠9/2⁠ ⁠7/2⁠ ⁠5/2⁠ ⁠3/2⁠ ⁠1/2⁠
⁠5/2⁠, −2 1 126 {\displaystyle {\sqrt {\frac {1}{126}}}} 4 63 {\displaystyle {\sqrt {\frac {4}{63}}}} 3 14 {\displaystyle {\sqrt {\frac {3}{14}}}} 8 21 {\displaystyle {\sqrt {\frac {8}{21}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}}
⁠3/2⁠, −1 10 63 {\displaystyle {\sqrt {\frac {10}{63}}}} 121 315 {\displaystyle {\sqrt {\frac {121}{315}}}} 6 35 {\displaystyle {\sqrt {\frac {6}{35}}}} − 2 105 {\displaystyle -{\sqrt {\frac {2}{105}}}} − 4 15 {\displaystyle -{\sqrt {\frac {4}{15}}}}
⁠1/2⁠, 0 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} 4 105 {\displaystyle {\sqrt {\frac {4}{105}}}} − 8 35 {\displaystyle -{\sqrt {\frac {8}{35}}}} − 2 35 {\displaystyle -{\sqrt {\frac {2}{35}}}} 1 5 {\displaystyle {\sqrt {\frac {1}{5}}}}
−⁠1/2⁠, 1 20 63 {\displaystyle {\sqrt {\frac {20}{63}}}} − 14 45 {\displaystyle -{\sqrt {\frac {14}{45}}}} 0 {\displaystyle 0} 5 21 {\displaystyle {\sqrt {\frac {5}{21}}}} − 2 15 {\displaystyle -{\sqrt {\frac {2}{15}}}}
−⁠3/2⁠, 2 5 126 {\displaystyle {\sqrt {\frac {5}{126}}}} − 64 315 {\displaystyle -{\sqrt {\frac {64}{315}}}} 27 70 {\displaystyle {\sqrt {\frac {27}{70}}}} − 32 105 {\displaystyle -{\sqrt {\frac {32}{105}}}} 1 15 {\displaystyle {\sqrt {\frac {1}{15}}}}

 j1 = ⁠5/2⁠,  j2 = ⁠5/2⁠

m = 5
j
m1, m2
 5
⁠5/2⁠, ⁠5/2⁠ 1 {\displaystyle 1}
m = 4
j
m1, m2
 5 4
⁠5/2⁠, ⁠3/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}}
⁠3/2⁠, ⁠5/2⁠ 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}}
m = 3
j
m1, m2
 5 4 3
⁠5/2⁠, ⁠1/2⁠ 2 9 {\displaystyle {\sqrt {\frac {2}{9}}}} 1 2 {\displaystyle {\sqrt {\frac {1}{2}}}} 5 18 {\displaystyle {\sqrt {\frac {5}{18}}}}
⁠3/2⁠, ⁠3/2⁠ 5 9 {\displaystyle {\sqrt {\frac {5}{9}}}} 0 {\displaystyle 0} − 2 3 {\displaystyle -{\frac {2}{3}}}
⁠1/2⁠, ⁠5/2⁠ 2 9 {\displaystyle {\sqrt {\frac {2}{9}}}} − 1 2 {\displaystyle -{\sqrt {\frac {1}{2}}}} 5 18 {\displaystyle {\sqrt {\frac {5}{18}}}}
m = 2
j
m1, m2
 5 4 3 2
⁠5/2⁠, −⁠1/2⁠ 1 12 {\displaystyle {\sqrt {\frac {1}{12}}}} 9 28 {\displaystyle {\sqrt {\frac {9}{28}}}} 5 12 {\displaystyle {\sqrt {\frac {5}{12}}}} 5 28 {\displaystyle {\sqrt {\frac {5}{28}}}}
⁠3/2⁠, ⁠1/2⁠ 5 12 {\displaystyle {\sqrt {\frac {5}{12}}}} 5 28 {\displaystyle {\sqrt {\frac {5}{28}}}} − 1 12 {\displaystyle -{\sqrt {\frac {1}{12}}}} − 9 28 {\displaystyle -{\sqrt {\frac {9}{28}}}}
⁠1/2⁠, ⁠3/2⁠ 5 12 {\displaystyle {\sqrt {\frac {5}{12}}}} − 5 28 {\displaystyle -{\sqrt {\frac {5}{28}}}} − 1 12 {\displaystyle -{\sqrt {\frac {1}{12}}}} 9 28 {\displaystyle {\sqrt {\frac {9}{28}}}}
−⁠1/2⁠, ⁠5/2⁠ 1 12 {\displaystyle {\sqrt {\frac {1}{12}}}} − 9 28 {\displaystyle -{\sqrt {\frac {9}{28}}}} 5 12 {\displaystyle {\sqrt {\frac {5}{12}}}} − 5 28 {\displaystyle -{\sqrt {\frac {5}{28}}}}
m = 1
j
m1, m2
 5 4 3 2 1
⁠5/2⁠, −⁠3/2⁠ 1 42 {\displaystyle {\sqrt {\frac {1}{42}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}}
⁠3/2⁠, −⁠1/2⁠ 5 21 {\displaystyle {\sqrt {\frac {5}{21}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 1 30 {\displaystyle {\sqrt {\frac {1}{30}}}} − 1 7 {\displaystyle -{\sqrt {\frac {1}{7}}}} − 8 35 {\displaystyle -{\sqrt {\frac {8}{35}}}}
⁠1/2⁠, ⁠1/2⁠ 10 21 {\displaystyle {\sqrt {\frac {10}{21}}}} 0 {\displaystyle 0} − 4 15 {\displaystyle -{\sqrt {\frac {4}{15}}}} 0 {\displaystyle 0} 9 35 {\displaystyle {\sqrt {\frac {9}{35}}}}
−⁠1/2⁠, ⁠3/2⁠ 5 21 {\displaystyle {\sqrt {\frac {5}{21}}}} − 5 14 {\displaystyle -{\sqrt {\frac {5}{14}}}} 1 30 {\displaystyle {\sqrt {\frac {1}{30}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}} − 8 35 {\displaystyle -{\sqrt {\frac {8}{35}}}}
−⁠3/2⁠, ⁠5/2⁠ 1 42 {\displaystyle {\sqrt {\frac {1}{42}}}} − 1 7 {\displaystyle -{\sqrt {\frac {1}{7}}}} 1 3 {\displaystyle {\sqrt {\frac {1}{3}}}} − 5 14 {\displaystyle -{\sqrt {\frac {5}{14}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}}
m = 0
j
m1, m2
 5 4 3 2 1 0
⁠5/2⁠, −⁠5/2⁠ 1 252 {\displaystyle {\sqrt {\frac {1}{252}}}} 1 28 {\displaystyle {\sqrt {\frac {1}{28}}}} 5 36 {\displaystyle {\sqrt {\frac {5}{36}}}} 25 84 {\displaystyle {\sqrt {\frac {25}{84}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}}
⁠3/2⁠, −⁠3/2⁠ 25 252 {\displaystyle {\sqrt {\frac {25}{252}}}} 9 28 {\displaystyle {\sqrt {\frac {9}{28}}}} 49 180 {\displaystyle {\sqrt {\frac {49}{180}}}} 1 84 {\displaystyle {\sqrt {\frac {1}{84}}}} − 9 70 {\displaystyle -{\sqrt {\frac {9}{70}}}} − 1 6 {\displaystyle -{\sqrt {\frac {1}{6}}}}
⁠1/2⁠, −⁠1/2⁠ 25 63 {\displaystyle {\sqrt {\frac {25}{63}}}} 1 7 {\displaystyle {\sqrt {\frac {1}{7}}}} − 4 45 {\displaystyle -{\sqrt {\frac {4}{45}}}} − 4 21 {\displaystyle -{\sqrt {\frac {4}{21}}}} 1 70 {\displaystyle {\sqrt {\frac {1}{70}}}} 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}}
−⁠1/2⁠, ⁠1/2⁠ 25 63 {\displaystyle {\sqrt {\frac {25}{63}}}} − 1 7 {\displaystyle -{\sqrt {\frac {1}{7}}}} − 4 45 {\displaystyle -{\sqrt {\frac {4}{45}}}} 4 21 {\displaystyle {\sqrt {\frac {4}{21}}}} 1 70 {\displaystyle {\sqrt {\frac {1}{70}}}} − 1 6 {\displaystyle -{\sqrt {\frac {1}{6}}}}
−⁠3/2⁠, ⁠3/2⁠ 25 252 {\displaystyle {\sqrt {\frac {25}{252}}}} − 9 28 {\displaystyle -{\sqrt {\frac {9}{28}}}} 49 180 {\displaystyle {\sqrt {\frac {49}{180}}}} − 1 84 {\displaystyle -{\sqrt {\frac {1}{84}}}} − 9 70 {\displaystyle -{\sqrt {\frac {9}{70}}}} 1 6 {\displaystyle {\sqrt {\frac {1}{6}}}}
−⁠5/2⁠, ⁠5/2⁠ 1 252 {\displaystyle {\sqrt {\frac {1}{252}}}} − 1 28 {\displaystyle -{\sqrt {\frac {1}{28}}}} 5 36 {\displaystyle {\sqrt {\frac {5}{36}}}} − 25 84 {\displaystyle -{\sqrt {\frac {25}{84}}}} 5 14 {\displaystyle {\sqrt {\frac {5}{14}}}} − 1 6 {\displaystyle -{\sqrt {\frac {1}{6}}}}

SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of j 1 {\displaystyle j_{1}} and j 2 {\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 (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.
  • 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