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Table of Clebsch–Gordan coefficients

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Summary

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:This article is incomplete due to technical limitations.

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{align} & \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{align}

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

For brevity, solutions with m 1 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.

{{math|size=100%|1= ''j''2 = 0}}

When , the Clebsch–Gordan coefficients are given by \delta_{j,j_1}\delta_{m,m_1}.

{{math|size=100%|1= ''j''1 = {{sfrac|1|2}},  ''j''2 = {{sfrac|1|2}}}}

1,
1
1−, −
1
10, −−,
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}

{{math|size=100%|1= ''j''1 = 1,  ''j''2 = 1/2}}

1,
1
1, −0,
\sqrt{\frac{1}{3}}\sqrt{\frac{2}{3}}
\sqrt{\frac{2}{3}}-\sqrt{\frac{1}{3}}

{{math|size=100%|1= ''j''1 = 1,  ''j''2 = 1}}

21, 1
1
211, 00, 1
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}
2101, −10, 0−1, 1
\sqrt{\frac{1}{6}}\sqrt{\frac{1}{2}}\sqrt{\frac{1}{3}}
\sqrt{\frac{2}{3}}0-\sqrt{\frac{1}{3}}
\sqrt{\frac{1}{6}}-\sqrt{\frac{1}{2}}\sqrt{\frac{1}{3}}

{{math|size=100%|1= ''j''1 = {{sfrac|3|2}},  ''j''2 = {{sfrac|1|2}}}}

2,
1
21, −,
\frac{1}{2}\sqrt{\frac{3}{4}}
\sqrt{\frac{3}{4}}-\frac{1}{2}
21, −−,
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}

{{math|size=100%|1= ''j''1 = {{sfrac|3|2}},  ''j''2 = 1}}

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

{{math|size=100%|1= ''j''1 = {{sfrac|3|2}},  ''j''2 = {{sfrac|3|2}}}}

3,
1
32,,
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}
321, −,−,
\sqrt{\frac{1}{5}}\sqrt{\frac{1}{2}}\sqrt{\frac{3}{10}}
\sqrt{\frac{3}{5}}0-\sqrt{\frac{2}{5}}
\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{2}}\sqrt{\frac{3}{10}}
3210, −, −−,−,
\sqrt{\frac{1}{20}}\frac{1}{2}\sqrt{\frac{9}{20}}\frac{1}{2}
\sqrt{\frac{9}{20}}\frac{1}{2}-\sqrt{\frac{1}{20}}-\frac{1}{2}
\sqrt{\frac{9}{20}}-\frac{1}{2}-\sqrt{\frac{1}{20}}\frac{1}{2}
\sqrt{\frac{1}{20}}-\frac{1}{2}\sqrt{\frac{9}{20}}-\frac{1}{2}

{{math|size=100%|1= ''j''1 = 2,  ''j''2 = {{sfrac|1|2}}}}

2,
1
2, −1,
\sqrt{\frac{1}{5}}\sqrt{\frac{4}{5}}
\sqrt{\frac{4}{5}}-\sqrt{\frac{1}{5}}
1, −0,
\sqrt{\frac{2}{5}}\sqrt{\frac{3}{5}}
\sqrt{\frac{3}{5}}-\sqrt{\frac{2}{5}}

{{math|size=100%|1= ''j''1 = 2,  ''j''2 = 1}}

32, 1
1
322, 01, 1
\sqrt{\frac{1}{3}}\sqrt{\frac{2}{3}}
\sqrt{\frac{2}{3}}-\sqrt{\frac{1}{3}}
3212, −11, 00, 1
\sqrt{\frac{1}{15}}\sqrt{\frac{1}{3}}\sqrt{\frac{3}{5}}
\sqrt{\frac{8}{15}}\sqrt{\frac{1}{6}}-\sqrt{\frac{3}{10}}
\sqrt{\frac{2}{5}}-\sqrt{\frac{1}{2}}\sqrt{\frac{1}{10}}
3211, −10, 0−1, 1
\sqrt{\frac{1}{5}}\sqrt{\frac{1}{2}}\sqrt{\frac{3}{10}}
\sqrt{\frac{3}{5}}0-\sqrt{\frac{2}{5}}
\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{2}}\sqrt{\frac{3}{10}}

{{math|size=100%|1= ''j''1 = 2,  ''j''2 = {{sfrac|3|2}}}}

2,
1
2,1,
\sqrt{\frac{3}{7}}\sqrt{\frac{4}{7}}
\sqrt{\frac{4}{7}}-\sqrt{\frac{3}{7}}
2, −1,0,
\sqrt{\frac{1}{7}}\sqrt{\frac{16}{35}}\sqrt{\frac{2}{5}}
\sqrt{\frac{4}{7}}\sqrt{\frac{1}{35}}-\sqrt{\frac{2}{5}}
\sqrt{\frac{2}{7}}-\sqrt{\frac{18}{35}}\sqrt{\frac{1}{5}}
2, −1, −0,−1,
\sqrt{\frac{1}{35}}\sqrt{\frac{6}{35}}\sqrt{\frac{2}{5}}\sqrt{\frac{2}{5}}
\sqrt{\frac{12}{35}}\sqrt{\frac{5}{14}}0-\sqrt{\frac{3}{10}}
\sqrt{\frac{18}{35}}-\sqrt{\frac{3}{35}}-\sqrt{\frac{1}{5}}\sqrt{\frac{1}{5}}
\sqrt{\frac{4}{35}}-\sqrt{\frac{27}{70}}\sqrt{\frac{2}{5}}-\sqrt{\frac{1}{10}}

{{math|size=100%|1= ''j''1 = 2,  ''j''2 = 2}}

42, 2
1
432, 11, 2
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}
4322, 01, 10, 2
\sqrt{\frac{3}{14}}\sqrt{\frac{1}{2}}\sqrt{\frac{2}{7}}
\sqrt{\frac{4}{7}}0-\sqrt{\frac{3}{7}}
\sqrt{\frac{3}{14}}-\sqrt{\frac{1}{2}}\sqrt{\frac{2}{7}}
43212, −11, 00, 1−1, 2
\sqrt{\frac{1}{14}}\sqrt{\frac{3}{10}}\sqrt{\frac{3}{7}}\sqrt{\frac{1}{5}}
\sqrt{\frac{3}{7}}\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{14}}-\sqrt{\frac{3}{10}}
\sqrt{\frac{3}{7}}-\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{14}}\sqrt{\frac{3}{10}}
\sqrt{\frac{1}{14}}-\sqrt{\frac{3}{10}}\sqrt{\frac{3}{7}}-\sqrt{\frac{1}{5}}
432102, −21, −10, 0−1, 1−2, 2
\sqrt{\frac{1}{70}}\sqrt{\frac{1}{10}}\sqrt{\frac{2}{7}}\sqrt{\frac{2}{5}}\sqrt{\frac{1}{5}}
\sqrt{\frac{8}{35}}\sqrt{\frac{2}{5}}\sqrt{\frac{1}{14}}-\sqrt{\frac{1}{10}}-\sqrt{\frac{1}{5}}
\sqrt{\frac{18}{35}}0-\sqrt{\frac{2}{7}}0\sqrt{\frac{1}{5}}
\sqrt{\frac{8}{35}}-\sqrt{\frac{2}{5}}\sqrt{\frac{1}{14}}\sqrt{\frac{1}{10}}-\sqrt{\frac{1}{5}}
\sqrt{\frac{1}{70}}-\sqrt{\frac{1}{10}}\sqrt{\frac{2}{7}}-\sqrt{\frac{2}{5}}\sqrt{\frac{1}{5}}

{{math|size=100%|1= ''j''1 = {{sfrac|5|2}},  ''j''2 = {{sfrac|1|2}}}}

3,
1
32, −,
\sqrt{\frac{1}{6}}\sqrt{\frac{5}{6}}
\sqrt{\frac{5}{6}}-\sqrt{\frac{1}{6}}
32, −,
\sqrt{\frac{1}{3}}\sqrt{\frac{2}{3}}
\sqrt{\frac{2}{3}}-\sqrt{\frac{1}{3}}
32, −−,
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}

{{math|size=100%|1= ''j''1 = {{sfrac|5|2}},  ''j''2 = 1}}

, 1
1
, 0, 1
\sqrt{\frac{2}{7}}\sqrt{\frac{5}{7}}
\sqrt{\frac{5}{7}}-\sqrt{\frac{2}{7}}
, −1, 0, 1
\sqrt{\frac{1}{21}}\sqrt{\frac{2}{7}}\sqrt{\frac{2}{3}}
\sqrt{\frac{10}{21}}\sqrt{\frac{9}{35}}-\sqrt{\frac{4}{15}}
\sqrt{\frac{10}{21}}-\sqrt{\frac{16}{35}}\sqrt{\frac{1}{15}}
, −1, 0−, 1
\sqrt{\frac{1}{7}}\sqrt{\frac{16}{35}}\sqrt{\frac{2}{5}}
\sqrt{\frac{4}{7}}\sqrt{\frac{1}{35}}-\sqrt{\frac{2}{5}}
\sqrt{\frac{2}{7}}-\sqrt{\frac{18}{35}}\sqrt{\frac{1}{5}}

{{math|size=100%|1= ''j''1 = {{sfrac|5|2}},  ''j''2 = {{sfrac|3|2}}}}

4,
1
43,,
\sqrt{\frac{3}{8}}\sqrt{\frac{5}{8}}
\sqrt{\frac{5}{8}}-\sqrt{\frac{3}{8}}
432, −,,
\sqrt{\frac{3}{28}}\sqrt{\frac{5}{12}}\sqrt{\frac{10}{21}}
\sqrt{\frac{15}{28}}\sqrt{\frac{1}{12}}-\sqrt{\frac{8}{21}}
\sqrt{\frac{5}{14}}-\sqrt{\frac{1}{2}}\sqrt{\frac{1}{7}}
4321, −, −,−,
\sqrt{\frac{1}{56}}\sqrt{\frac{1}{8}}\sqrt{\frac{5}{14}}\sqrt{\frac{1}{2}}
\sqrt{\frac{15}{56}}\sqrt{\frac{49}{120}}\sqrt{\frac{1}{42}}-\sqrt{\frac{3}{10}}
\sqrt{\frac{15}{28}}-\sqrt{\frac{1}{60}}-\sqrt{\frac{25}{84}}\sqrt{\frac{3}{20}}
\sqrt{\frac{5}{28}}-\sqrt{\frac{9}{20}}\sqrt{\frac{9}{28}}-\sqrt{\frac{1}{20}}
4321, −, −−,−,
\sqrt{\frac{1}{14}}\sqrt{\frac{3}{10}}\sqrt{\frac{3}{7}}\sqrt{\frac{1}{5}}
\sqrt{\frac{3}{7}}\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{14}}-\sqrt{\frac{3}{10}}
\sqrt{\frac{3}{7}}-\sqrt{\frac{1}{5}}-\sqrt{\frac{1}{14}}\sqrt{\frac{3}{10}}
\sqrt{\frac{1}{14}}-\sqrt{\frac{3}{10}}\sqrt{\frac{3}{7}}-\sqrt{\frac{1}{5}}

{{math|size=100%|1= ''j''1 = {{sfrac|5|2}},  ''j''2 = 2}}

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

{{math|size=100%|1= ''j''1 = {{sfrac|5|2}},  ''j''2 = {{sfrac|5|2}}}}

5,
1
54,,
\sqrt{\frac{1}{2}}\sqrt{\frac{1}{2}}
\sqrt{\frac{1}{2}}-\sqrt{\frac{1}{2}}
543,,,
\sqrt{\frac{2}{9}}\sqrt{\frac{1}{2}}\sqrt{\frac{5}{18}}
\sqrt{\frac{5}{9}}0-{\frac{2}{3}}
\sqrt{\frac{2}{9}}-\sqrt{\frac{1}{2}}\sqrt{\frac{5}{18}}
5432, −,,−,
\sqrt{\frac{1}{12}}\sqrt{\frac{9}{28}}\sqrt{\frac{5}{12}}\sqrt{\frac{5}{28}}
\sqrt{\frac{5}{12}}\sqrt{\frac{5}{28}}-\sqrt{\frac{1}{12}}-\sqrt{\frac{9}{28}}
\sqrt{\frac{5}{12}}-\sqrt{\frac{5}{28}}-\sqrt{\frac{1}{12}}\sqrt{\frac{9}{28}}
\sqrt{\frac{1}{12}}-\sqrt{\frac{9}{28}}\sqrt{\frac{5}{12}}-\sqrt{\frac{5}{28}}
54321, −, −,−,−,
\sqrt{\frac{1}{42}}\sqrt{\frac{1}{7}}\sqrt{\frac{1}{3}}\sqrt{\frac{5}{14}}\sqrt{\frac{1}{7}}
\sqrt{\frac{5}{21}}\sqrt{\frac{5}{14}}\sqrt{\frac{1}{30}}-\sqrt{\frac{1}{7}}-\sqrt{\frac{8}{35}}
\sqrt{\frac{10}{21}}0-\sqrt{\frac{4}{15}}0\sqrt{\frac{9}{35}}
\sqrt{\frac{5}{21}}-\sqrt{\frac{5}{14}}\sqrt{\frac{1}{30}}\sqrt{\frac{1}{7}}-\sqrt{\frac{8}{35}}
\sqrt{\frac{1}{42}}-\sqrt{\frac{1}{7}}\sqrt{\frac{1}{3}}-\sqrt{\frac{5}{14}}\sqrt{\frac{1}{7}}
543210, −, −, −−,−,−,
\sqrt{\frac{1}{252}}\sqrt{\frac{1}{28}}\sqrt{\frac{5}{36}}\sqrt{\frac{25}{84}}\sqrt{\frac{5}{14}}\sqrt{\frac{1}{6}}
\sqrt{\frac{25}{252}}\sqrt{\frac{9}{28}}\sqrt{\frac{49}{180}}\sqrt{\frac{1}{84}}-\sqrt{\frac{9}{70}}-\sqrt{\frac{1}{6}}
\sqrt{\frac{25}{63}}\sqrt{\frac{1}{7}}-\sqrt{\frac{4}{45}}-\sqrt{\frac{4}{21}}\sqrt{\frac{1}{70}}\sqrt{\frac{1}{6}}
\sqrt{\frac{25}{63}}-\sqrt{\frac{1}{7}}-\sqrt{\frac{4}{45}}\sqrt{\frac{4}{21}}\sqrt{\frac{1}{70}}-\sqrt{\frac{1}{6}}
\sqrt{\frac{25}{252}}-\sqrt{\frac{9}{28}}\sqrt{\frac{49}{180}}-\sqrt{\frac{1}{84}}-\sqrt{\frac{9}{70}}\sqrt{\frac{1}{6}}
\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

References

  1. Baird, C.E.. (October 1964). "On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU''n''". J. Math. Phys..
  2. Hagiwara, K.. (July 2002). "Review of Particle Properties". Phys. Rev. D.
  3. Mathar, Richard J.. (2006-08-14). "SO(3) Clebsch Gordan coefficients".
  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.
  6. Alex, A.. (February 2011). "A numerical algorithm for the explicit calculation of SU(N) and SL(N,C) Clebsch–Gordan coefficients". J. Math. Phys..
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