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Table of Clebsch–Gordan coefficients

From Wikipedia, the free encyclopedia

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 , , 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

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The Clebsch–Gordan coefficients are the solutions to

Explicitly:

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

and

Specific values

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

 j2 = 0

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When j2 = 0, the Clebsch–Gordan coefficients are given by .

 j1 = 1/2j2 = 1/2

[edit]
m = 1
j
m1m2
1
1/21/2
m = −1
j
m1m2
1
1/2, −1/2
m = 0
j
m1m2
1 0
1/2, −1/2
1/21/2

 j1 = 1,  j2 = 1/2

[edit]
m = 3/2
j
m1m2
3/2
1, 1/2
m = 1/2
j
m1m2
3/2 1/2
1, −1/2
0, 1/2

 j1 = 1,  j2 = 1

[edit]
m = 2
j
m1m2
2
1, 1
m = 1
j
m1m2
2 1
1, 0
0, 1
m = 0
j
m1m2
2 1 0
1, −1
0, 0
−1, 1

 j1 = 3/2j2 = 1/2

[edit]
m = 2
j
m1m2
2
3/21/2
m = 1
j
m1m2
2 1
3/2, −1/2
1/21/2
m = 0
j
m1m2
2 1
1/2, −1/2
1/21/2

 j1 = 3/2j2 = 1

[edit]
m = 5/2
j
m1m2
5/2
3/2, 1
m = 3/2
j
m1m2
5/2 3/2
3/2, 0
1/2, 1
m = 1/2
j
m1m2
5/2 3/2 1/2
3/2, −1
1/2, 0
1/2, 1

 j1 = 3/2j2 = 3/2

[edit]
m = 3
j
m1m2
3
3/23/2
m = 2
j
m1m2
3 2
3/21/2
1/23/2
m = 1
j
m1m2
3 2 1
3/2, −1/2
1/21/2
1/23/2
m = 0
j
m1m2
3 2 1 0
3/2, −3/2
1/2, −1/2
1/21/2
3/23/2

 j1 = 2,  j2 = 1/2

[edit]
m = 5/2
j
m1m2
5/2
2, 1/2
m = 3/2
j
m1m2
5/2 3/2
2, −1/2
1, 1/2
m = 1/2
j
m1m2
5/2 3/2
1, −1/2
0, 1/2

 j1 = 2,  j2 = 1

[edit]
m = 3
j
m1m2
3
2, 1
m = 2
j
m1m2
3 2
2, 0
1, 1
m = 1
j
m1m2
3 2 1
2, −1
1, 0
0, 1
m = 0
j
m1m2
3 2 1
1, −1
0, 0
−1, 1

 j1 = 2,  j2 = 3/2

[edit]
m = 7/2
j
m1m2
7/2
2, 3/2
m = 5/2
j
m1m2
7/2 5/2
2, 1/2
1, 3/2
m = 3/2
j
m1m2
7/2 5/2 3/2
2, −1/2
1, 1/2
0, 3/2
m = 1/2
j
m1m2
7/2 5/2 3/2 1/2
2, −3/2
1, −1/2
0, 1/2
−1, 3/2

 j1 = 2,  j2 = 2

[edit]
m = 4
j
m1m2
4
2, 2
m = 3
j
m1m2
4 3
2, 1
1, 2
m = 2
j
m1m2
4 3 2
2, 0
1, 1
0, 2
m = 1
j
m1m2
4 3 2 1
2, −1
1, 0
0, 1
−1, 2
m = 0
j
m1m2
4 3 2 1 0
2, −2
1, −1
0, 0
−1, 1
−2, 2

 j1 = 5/2j2 = 1/2

[edit]
m = 3
j
m1m2
3
5/21/2
m = 2
j
m1m2
3 2
5/2, −1/2
3/21/2
m = 1
j
m1m2
3 2
3/2, −1/2
1/21/2
m = 0
j
m1m2
3 2
1/2, −1/2
1/21/2

 j1 = 5/2j2 = 1

[edit]
m = 7/2
j
m1m2
7/2
5/2, 1
m = 5/2
j
m1m2
7/2 5/2
5/2, 0
3/2, 1
m = 3/2
j
m1m2
7/2 5/2 3/2
5/2, −1
3/2, 0
1/2, 1
m = 1/2
j
m1m2
7/2 5/2 3/2
3/2, −1
1/2, 0
1/2, 1

 j1 = 5/2j2 = 3/2

[edit]
m = 4
j
m1m2
4
5/23/2
m = 3
j
m1m2
4 3
5/21/2
3/23/2
m = 2
j
m1m2
4 3 2
5/2, −1/2
3/21/2
1/23/2
m = 1
j
m1m2
4 3 2 1
5/2, −3/2
3/2, −1/2
1/21/2
1/23/2
m = 0
j
m1m2
4 3 2 1
3/2, −3/2
1/2, −1/2
1/21/2
3/23/2

 j1 = 5/2j2 = 2

[edit]
m = 9/2
j
m1m2
9/2
5/2, 2
m = 7/2
j
m1m2
9/2 7/2
5/2, 1
3/2, 2
m = 5/2
j
m1m2
9/2 7/2 5/2
5/2, 0
3/2, 1
1/2, 2
m = 3/2
j
m1m2
9/2 7/2 5/2 3/2
5/2, −1
3/2, 0
1/2, 1
1/2, 2
m = 1/2
j
m1m2
9/2 7/2 5/2 3/2 1/2
5/2, −2
3/2, −1
1/2, 0
1/2, 1
3/2, 2

 j1 = 5/2j2 = 5/2

[edit]
m = 5
j
m1m2
5
5/25/2
m = 4
j
m1m2
5 4
5/23/2
3/25/2
m = 3
j
m1m2
5 4 3
5/21/2
3/23/2
1/25/2
m = 2
j
m1m2
5 4 3 2
5/2, −1/2
3/21/2
1/23/2
1/25/2
m = 1
j
m1m2
5 4 3 2 1
5/2, −3/2
3/2, −1/2
1/21/2
1/23/2
3/25/2
m = 0
j
m1m2
5 4 3 2 1 0
5/2, −5/2
3/2, −3/2
1/2, −1/2
1/21/2
3/23/2
5/25/2

SU(N) Clebsch–Gordan coefficients

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Algorithms to produce Clebsch–Gordan coefficients for higher values of and , 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

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  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.
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