Table of Clebsch–Gordan coefficients explained

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

j1

,

j2

,

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

|j1,j2;J,M\rangle=

j1
\sum
m1=-j1
j2
\sum
m2=-j2

|j1,m1;j2,m2\rangle\langlej1,j2;m1,m2\midj1,j2;J,M\rangle

Explicitly:

\begin{align} &\langlej1,j2;m1,m2\midj1,j2;J,M\rangle\\[6pt] ={}&

\delta\sqrt{
M,m1+m2
(2J+1)(J+j1-j2)!(J-j1+j2)!(j1+j2-J)!
(j1+j2+J+1)!
}\ \times \\[6pt]&\sqrt\ \times \\[6pt]&\sum_k \frac.\end

The summation is extended over all integer for which the argument of every factorial is nonnegative.[4]

For brevity, solutions with and are omitted. They may be calculated using the simple relations

\langlej1,j2;m1,m2\midj1,j

J-j1-j2
2;J,M\rangle=(-1)

\langlej1,j2;-m1,-m2\midj1,j2;J,-M\rangle.

and

\langlej1,j2;m1,m2\midj1,j

J-j1-j2
2;J,M\rangle=(-1)

\langlej2,j1;m2,m1\midj2,j1;J,M\rangle.

Specific values

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

When, the Clebsch–Gordan coefficients are given by

\delta
j,j1
\delta
m,m1
.
1 0
, −
\sqrt{1
2
}
\sqrt{1
2
}
−, 
\sqrt{1
2
}
-\sqrt{1
2
}
1, −
\sqrt{1
3
}
\sqrt{2
3
}
0, 
\sqrt{2
3
}
-\sqrt{1
3
}
2 1
1, 0
\sqrt{1
2
}
\sqrt{1
2
}
0, 1
\sqrt{1
2
}
-\sqrt{1
2
}
2 1 0
1, −1
\sqrt{1
6
}
\sqrt{1
2
}
\sqrt{1
3
}
0, 0
\sqrt{2
3
}

0

-\sqrt{1
3
}
−1, 1
\sqrt{1
6
}
-\sqrt{1
2
}
\sqrt{1
3
}
2 1
, −
1
2
\sqrt{3
4
}
, 
\sqrt{3
4
}
-1
2
2 1
, −
\sqrt{1
2
}
\sqrt{1
2
}
−, 
\sqrt{1
2
}
-\sqrt{1
2
}
, 0
\sqrt{2
5
}
\sqrt{3
5
}
, 1
\sqrt{3
5
}
-\sqrt{2
5
}
, −1
\sqrt{1
10
}
\sqrt{2
5
}
\sqrt{1
2
}
, 0
\sqrt{3
5
}
\sqrt{1
15
}
-\sqrt{1
3
}
−, 1
\sqrt{3
10
}
-\sqrt{8
15
}
\sqrt{1
6
}
3 2
, 
\sqrt{1
2
}
\sqrt{1
2
}
, 
\sqrt{1
2
}
-\sqrt{1
2
}
3 2 1
, −
\sqrt{1
5
}
\sqrt{1
2
}
\sqrt{3
10
}
, 
\sqrt{3
5
}

0

-\sqrt{2
5
}
−, 
\sqrt{1
5
}
-\sqrt{1
2
}
\sqrt{3
10
}
3 2 1 0
, −
\sqrt{1
20
}
1
2
\sqrt{9
20
}
1
2
, −
\sqrt{9
20
}
1
2
-\sqrt{1
20
}
-1
2
−, 
\sqrt{9
20
}
-1
2
-\sqrt{1
20
}
1
2
−, 
\sqrt{1
20
}
-1
2
\sqrt{9
20
}
-1
2
2, −
\sqrt{1
5
}
\sqrt{4
5
}
1, 
\sqrt{4
5
}
-\sqrt{1
5
}
1, −
\sqrt{2
5
}
\sqrt{3
5
}
0, 
\sqrt{3
5
}
-\sqrt{2
5
}
3 2
2, 0
\sqrt{1
3
}
\sqrt{2
3
}
1, 1
\sqrt{2
3
}
-\sqrt{1
3
}
3 2 1
2, −1
\sqrt{1
15
}
\sqrt{1
3
}
\sqrt{3
5
}
1, 0
\sqrt{8
15
}
\sqrt{1
6
}
-\sqrt{3
10
}
0, 1
\sqrt{2
5
}
-\sqrt{1
2
}
\sqrt{1
10
}
3 2 1
1, −1
\sqrt{1
5
}
\sqrt{1
2
}
\sqrt{3
10
}
0, 0
\sqrt{3
5
}

0

-\sqrt{2
5
}
−1, 1
\sqrt{1
5
}
-\sqrt{1
2
}
\sqrt{3
10
}
2, 
\sqrt{3
7
}
\sqrt{4
7
}
1, 
\sqrt{4
7
}
-\sqrt{3
7
}
2, −
\sqrt{1
7
}
\sqrt{16
35
}
\sqrt{2
5
}
1, 
\sqrt{4
7
}
\sqrt{1
35
}
-\sqrt{2
5
}
0, 
\sqrt{2
7
}
-\sqrt{18
35
}
\sqrt{1
5
}
2, −
\sqrt{1
35
}
\sqrt{6
35
}
\sqrt{2
5
}
\sqrt{2
5
}
1, −
\sqrt{12
35
}
\sqrt{5
14
}

0

-\sqrt{3
10
}
0, 
\sqrt{18
35
}
-\sqrt{3
35
}
-\sqrt{1
5
}
\sqrt{1
5
}
−1, 
\sqrt{4
35
}
-\sqrt{27
70
}
\sqrt{2
5
}
-\sqrt{1
10
}
4 3
2, 1
\sqrt{1
2
}
\sqrt{1
2
}
1, 2
\sqrt{1
2
}
-\sqrt{1
2
}
4 3 2
2, 0
\sqrt{3
14
}
\sqrt{1
2
}
\sqrt{2
7
}
1, 1
\sqrt{4
7
}

0

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

0

-\sqrt{2
7
}

0

\sqrt{1
5
}
−1, 1
\sqrt{8
35
}
-\sqrt{2
5
}
\sqrt{1
14
}
\sqrt{1
10
}
-\sqrt{1
5
}
−2, 2
\sqrt{1
70
}
-\sqrt{1
10
}
\sqrt{2
7
}
-\sqrt{2
5
}
\sqrt{1
5
}
3 2
, −
\sqrt{1
6
}
\sqrt{5
6
}
, 
\sqrt{5
6
}
-\sqrt{1
6
}
3 2
, −
\sqrt{1
3
}
\sqrt{2
3
}
, 
\sqrt{2
3
}
-\sqrt{1
3
}
3 2
, −
\sqrt{1
2
}
\sqrt{1
2
}
−, 
\sqrt{1
2
}
-\sqrt{1
2
}
, 0
\sqrt{2
7
}
\sqrt{5
7
}
, 1
\sqrt{5
7
}
-\sqrt{2
7
}
, −1
\sqrt{1
21
}
\sqrt{2
7
}
\sqrt{2
3
}
, 0
\sqrt{10
21
}
\sqrt{9
35
}
-\sqrt{4
15
}
, 1
\sqrt{10
21
}
-\sqrt{16
35
}
\sqrt{1
15
}
, −1
\sqrt{1
7
}
\sqrt{16
35
}
\sqrt{2
5
}
, 0
\sqrt{4
7
}
\sqrt{1
35
}
-\sqrt{2
5
}
−, 1
\sqrt{2
7
}
-\sqrt{18
35
}
\sqrt{1
5
}
4 3
, 
\sqrt{3
8
}
\sqrt{5
8
}
, 
\sqrt{5
8
}
-\sqrt{3
8
}
4 3 2
, −
\sqrt{3
28
}
\sqrt{5
12
}
\sqrt{10
21
}
, 
\sqrt{15
28
}
\sqrt{1
12
}
-\sqrt{8
21
}
, 
\sqrt{5
14
}
-\sqrt{1
2
}
\sqrt{1
7
}
4 3 2 1
, −
\sqrt{1
56
}
\sqrt{1
8
}
\sqrt{5
14
}
\sqrt{1
2
}
, −
\sqrt{15
56
}
\sqrt{49
120
}
\sqrt{1
42
}
-\sqrt{3
10
}
, 
\sqrt{15
28
}
-\sqrt{1
60
}
-\sqrt{25
84
}
\sqrt{3
20
}
−, 
\sqrt{5
28
}
-\sqrt{9
20
}
\sqrt{9
28
}
-\sqrt{1
20
}
4 3 2 1
, −
\sqrt{1
14
}
\sqrt{3
10
}
\sqrt{3
7
}
\sqrt{1
5
}
, −
\sqrt{3
7
}
\sqrt{1
5
}
-\sqrt{1
14
}
-\sqrt{3
10
}
−, 
\sqrt{3
7
}
-\sqrt{1
5
}
-\sqrt{1
14
}
\sqrt{3
10
}
−, 
\sqrt{1
14
}
-\sqrt{3
10
}
\sqrt{3
7
}
-\sqrt{1
5
}
, 1
2
3
\sqrt{5
9
}
, 2
\sqrt{5
9
}
-2
3
, 0
\sqrt{1
6
}
\sqrt{10
21
}
\sqrt{5
14
}
, 1
\sqrt{5
9
}
\sqrt{1
63
}
-\sqrt{3
7
}
, 2
\sqrt{5
18
}
-\sqrt{32
63
}
\sqrt{3
14
}
, −1
\sqrt{1
21
}
\sqrt{5
21
}
\sqrt{3
7
}
\sqrt{2
7
}
, 0
\sqrt{5
14
}
\sqrt{2
7
}
-\sqrt{1
70
}
-\sqrt{12
35
}
, 1
\sqrt{10
21
}
-\sqrt{2
21
}
-\sqrt{6
35
}
\sqrt{9
35
}
−, 2
\sqrt{5
42
}
-\sqrt{8
21
}
\sqrt{27
70
}
-\sqrt{4
35
}
, −2
\sqrt{1
126
}
\sqrt{4
63
}
\sqrt{3
14
}
\sqrt{8
21
}
\sqrt{1
3
}
, −1
\sqrt{10
63
}
\sqrt{121
315
}
\sqrt{6
35
}
-\sqrt{2
105
}
-\sqrt{4
15
}
, 0
\sqrt{10
21
}
\sqrt{4
105
}
-\sqrt{8
35
}
-\sqrt{2
35
}
\sqrt{1
5
}
−, 1
\sqrt{20
63
}
-\sqrt{14
45
}

0

\sqrt{5
21
}
-\sqrt{2
15
}
−, 2
\sqrt{5
126
}
-\sqrt{64
315
}
\sqrt{27
70
}
-\sqrt{32
105
}
\sqrt{1
15
}
54
, 
\sqrt{1
2
}
\sqrt{1
2
}
, 
\sqrt{1
2
}
-\sqrt{1
2
}
543
, 
\sqrt{2
9
}
\sqrt{1
2
}
\sqrt{5
18
}
, 
\sqrt{5
9
}

0

-{2
3
}
, 
\sqrt{2
9
}
-\sqrt{1
2
}
\sqrt{5
18
}
5432
, −
\sqrt{1
12
}
\sqrt{9
28
}
\sqrt{5
12
}
\sqrt{5
28
}
, 
\sqrt{5
12
}
\sqrt{5
28
}
-\sqrt{1
12
}
-\sqrt{9
28
}
, 
\sqrt{5
12
}
-\sqrt{5
28
}
-\sqrt{1
12
}
\sqrt{9
28
}
−, 
\sqrt{1
12
}
-\sqrt{9
28
}
\sqrt{5
12
}
-\sqrt{5
28
}
54 3 21
, −
\sqrt{1
42
}
\sqrt{1
7
}
\sqrt{1
3
}
\sqrt{5
14
}
\sqrt{1
7
}
, −
\sqrt{5
21
}
\sqrt{5
14
}
\sqrt{1
30
}
-\sqrt{1
7
}
-\sqrt{8
35
}
, 
\sqrt{10
21
}

0

-\sqrt{4
15
}

0

\sqrt{9
35
}
−, 
\sqrt{5
21
}
-\sqrt{5
14
}
\sqrt{1
30
}
\sqrt{1
7
}
-\sqrt{8
35
}
−, 
\sqrt{1
42
}
-\sqrt{1
7
}
\sqrt{1
3
}
-\sqrt{5
14
}
\sqrt{1
7
}
54 3 210
, −
\sqrt{1
252
}
\sqrt{1
28
}
\sqrt{5
36
}
\sqrt{25
84
}
\sqrt{5
14
}
\sqrt{1
6
}
, −
\sqrt{25
252
}
\sqrt{9
28
}
\sqrt{49
180
}
\sqrt{1
84
}
-\sqrt{9
70
}
-\sqrt{1
6
}
, −
\sqrt{25
63
}
\sqrt{1
7
}
-\sqrt{4
45
}
-\sqrt{4
21
}
\sqrt{1
70
}
\sqrt{1
6
}
−, 
\sqrt{25
63
}
-\sqrt{1
7
}
-\sqrt{4
45
}
\sqrt{4
21
}
\sqrt{1
70
}
-\sqrt{1
6
}
−, 
\sqrt{25
252
}
-\sqrt{9
28
}
\sqrt{49
180
}
-\sqrt{1
84
}
-\sqrt{9
70
}
\sqrt{1
6
}
−, 
\sqrt{1
252
}
-\sqrt{1
28
}
\sqrt{5
36
}
-\sqrt{25
84
}
\sqrt{5
14
}
-\sqrt{1
6
}

SU(N) Clebsch–Gordan coefficients

Algorithms to produce Clebsch–Gordan coefficients for higher values of

j1

and

j2

, 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 . On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SUn . J. Math. Phys. . 5 . 12 . October 1964 . 1723–1730 . 10.1063/1.1704095 . 1964JMP.....5.1723B.
  2. Hagiwara . K. . Review of Particle Properties . Phys. Rev. D . 66 . 1 . July 2002 . 010001 . 10.1103/PhysRevD.66.010001 . 2007-12-20 . 2002PhRvD..66a0001H. etal.
  3. Web site: Mathar . Richard J. . SO(3) Clebsch Gordan coefficients . 2006-08-14 . text . 2012-10-15.
  4. (2.41), p. 172, Quantum Mechanics: Foundations and Applications, Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, .
  5. Book: Weissbluth, Mitchel . Atoms and molecules . registration . 1978 . ACADEMIC PRESS . 0-12-744450-5 . 28. Table 1.4 resumes the most common.
  6. Alex . A. . M. Kalus . A. Huckleberry . J. von Delft . A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients . J. Math. Phys. . 82 . February 2011 . 023507 . 10.1063/1.3521562 . 2011JMP....52b3507A . 1009.0437 .

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