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_1j_2)jm\rangle = \sum_{m_1=-j_1}^{j_1} \sum_{m_2=-j_2}^{j_2} |j_1m_1j_2m_2\rangle \langle j_1j_2;m_1m_2|j_1j_2;jm\rangle

Explicitly:

\begin{align} \langle j_1j_2;m_1m_2|j_1j_2;jm\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{align}

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

\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2}\langle j_1j_2;-m_1,-m_2|j_1j_2;j,-m\rangle .

and

\langle j_1j_2;m_1m_2|j_1j_2;jm\rangle=(-1)^{j-j_1-j_2} \langle j_2j_1;m_2m_1|j_2j_1;jm\rangle .


A complete list [5]

 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 j
m1, m2
1
1/2, 1/2 1\!\,
m=0 j
m1, 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 j
m1, m2
3/2
1, 1/2 1\!\,
m=1/2 j
m1, 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 j
m1, m2
2
1, 1 1\!\,
m=1 j
m1, 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 j
m1, 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 j
m1, m2
2
3/2, 1/2 1\!\,
m=1 j
m1, 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 j
m1, 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 j
m1, m2
5/2
3/2, 1 1\!\,
m=3/2 j
m1, 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 j
m1, 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 j
m1, m2
3
3/2, 3/2 1\!\,
m=2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
5/2
2, 1/2 1\!\,
m=3/2 j
m1, 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 j
m1, 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 j
m1, m2
3
2, 1 1\!\,
m=2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
7/2
2, 3/2 1\!\,
m=5/2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
4
2, 2 1\!\,
m=3 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
3
5/2, 1/2 1\!\,
m=2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
7/2
5/2, 1 1\!\,
m=5/2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
4
5/2, 3/2 1\!\,
m=3 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, m2
9/2
5/2, 2 1\!\,
m=7/2 j
m1, 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 j
m1, 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 j
m1, 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 j
m1, 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}}\!\,

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

  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: 1723–1730. Bibcode:1964JMP.....5.1723B. doi:10.1063/1.1704095. Retrieved 2007-12-20.
  2. ↑ Hagiwara, K.; et al. (July 2002). "Review of Particle Properties" (PDF). Phys. Rev. D 66: 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. ↑ Weisbluth, Michael (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. Retrieved 2011-04-13.

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