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 , , 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
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
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 .
j1 = 1/2, j2 = 1/2
j m1, m2 | 1 |
---|---|
1/2, 1/2 |
j m1, m2 | 1 |
---|---|
−1/2, −1/2 |
j m1, m2 | 1 | 0 |
---|---|---|
1/2, −1/2 | ||
−1/2, 1/2 |
j1 = 1, j2 = 1/2
j m1, m2 | 3/2 |
---|---|
1, 1/2 |
j m1, m2 | 3/2 | 1/2 |
---|---|---|
1, −1/2 | ||
0, 1/2 |
j1 = 1, j2 = 1
j m1, m2 | 2 |
---|---|
1, 1 |
j m1, m2 | 2 | 1 |
---|---|---|
1, 0 | ||
0, 1 |
j m1, m2 | 2 | 1 | 0 |
---|---|---|---|
1, −1 | |||
0, 0 | |||
−1, 1 |
j1 = 3/2, j2 = 1/2
j m1, m2 | 2 |
---|---|
3/2, 1/2 |
j m1, m2 | 2 | 1 |
---|---|---|
3/2, −1/2 | ||
1/2, 1/2 |
j m1, m2 | 2 | 1 |
---|---|---|
1/2, −1/2 | ||
−1/2, 1/2 |
j1 = 3/2, j2 = 1
j m1, m2 | 5/2 |
---|---|
3/2, 1 |
j m1, m2 | 5/2 | 3/2 |
---|---|---|
3/2, 0 | ||
1/2, 1 |
j m1, m2 | 5/2 | 3/2 | 1/2 |
---|---|---|---|
3/2, −1 | |||
1/2, 0 | |||
−1/2, 1 |
j1 = 3/2, j2 = 3/2
j m1, m2 | 3 |
---|---|
3/2, 3/2 |
j m1, m2 | 3 | 2 |
---|---|---|
3/2, 1/2 | ||
1/2, 3/2 |
j m1, m2 | 3 | 2 | 1 |
---|---|---|---|
3/2, −1/2 | |||
1/2, 1/2 | |||
−1/2, 3/2 |
j m1, m2 | 3 | 2 | 1 | 0 |
---|---|---|---|---|
3/2, −3/2 | ||||
1/2, −1/2 | ||||
−1/2, 1/2 | ||||
−3/2, 3/2 |
j1 = 2, j2 = 1/2
j m1, m2 | 5/2 |
---|---|
2, 1/2 |
j m1, m2 | 5/2 | 3/2 |
---|---|---|
2, −1/2 | ||
1, 1/2 |
j m1, m2 | 5/2 | 3/2 |
---|---|---|
1, −1/2 | ||
0, 1/2 |
j1 = 2, j2 = 1
j m1, m2 | 3 |
---|---|
2, 1 |
j m1, m2 | 3 | 2 |
---|---|---|
2, 0 | ||
1, 1 |
j m1, m2 | 3 | 2 | 1 |
---|---|---|---|
2, −1 | |||
1, 0 | |||
0, 1 |
j m1, m2 | 3 | 2 | 1 |
---|---|---|---|
1, −1 | |||
0, 0 | |||
−1, 1 |
j1 = 2, j2 = 3/2
j m1, m2 | 7/2 |
---|---|
2, 3/2 |
j m1, m2 | 7/2 | 5/2 |
---|---|---|
2, 1/2 | ||
1, 3/2 |
j m1, m2 | 7/2 | 5/2 | 3/2 |
---|---|---|---|
2, −1/2 | |||
1, 1/2 | |||
0, 3/2 |
j m1, m2 | 7/2 | 5/2 | 3/2 | 1/2 |
---|---|---|---|---|
2, −3/2 | ||||
1, −1/2 | ||||
0, 1/2 | ||||
−1, 3/2 |
j1 = 2, j2 = 2
j m1, m2 | 4 |
---|---|
2, 2 |
j m1, m2 | 4 | 3 |
---|---|---|
2, 1 | ||
1, 2 |
j m1, m2 | 4 | 3 | 2 |
---|---|---|---|
2, 0 | |||
1, 1 | |||
0, 2 |
j m1, m2 | 4 | 3 | 2 | 1 |
---|---|---|---|---|
2, −1 | ||||
1, 0 | ||||
0, 1 | ||||
−1, 2 |
j m1, m2 | 4 | 3 | 2 | 1 | 0 |
---|---|---|---|---|---|
2, −2 | |||||
1, −1 | |||||
0, 0 | |||||
−1, 1 | |||||
−2, 2 |
j1 = 5/2, j2 = 1/2
j m1, m2 | 3 |
---|---|
5/2, 1/2 |
j m1, m2 | 3 | 2 |
---|---|---|
5/2, −1/2 | ||
3/2, 1/2 |
j m1, m2 | 3 | 2 |
---|---|---|
3/2, −1/2 | ||
1/2, 1/2 |
j m1, m2 | 3 | 2 |
---|---|---|
1/2, −1/2 | ||
−1/2, 1/2 |
j1 = 5/2, j2 = 1
j m1, m2 | 7/2 |
---|---|
5/2, 1 |
j m1, m2 | 7/2 | 5/2 |
---|---|---|
5/2, 0 | ||
3/2, 1 |
j m1, m2 | 7/2 | 5/2 | 3/2 |
---|---|---|---|
5/2, −1 | |||
3/2, 0 | |||
1/2, 1 |
j m1, m2 | 7/2 | 5/2 | 3/2 |
---|---|---|---|
3/2, −1 | |||
1/2, 0 | |||
−1/2, 1 |
j1 = 5/2, j2 = 3/2
j m1, m2 | 4 |
---|---|
5/2, 3/2 |
j m1, m2 | 4 | 3 |
---|---|---|
5/2, 1/2 | ||
3/2, 3/2 |
j m1, m2 | 4 | 3 | 2 |
---|---|---|---|
5/2, −1/2 | |||
3/2, 1/2 | |||
1/2, 3/2 |
j m1, m2 | 4 | 3 | 2 | 1 |
---|---|---|---|---|
5/2, −3/2 | ||||
3/2, −1/2 | ||||
1/2, 1/2 | ||||
−1/2, 3/2 |
j m1, m2 | 4 | 3 | 2 | 1 |
---|---|---|---|---|
3/2, −3/2 | ||||
1/2, −1/2 | ||||
−1/2, 1/2 | ||||
−3/2, 3/2 |
j1 = 5/2, j2 = 2
j m1, m2 | 9/2 |
---|---|
5/2, 2 |
j m1, m2 | 9/2 | 7/2 |
---|---|---|
5/2, 1 | ||
3/2, 2 |
j m1, m2 | 9/2 | 7/2 | 5/2 |
---|---|---|---|
5/2, 0 | |||
3/2, 1 | |||
1/2, 2 |
j m1, m2 | 9/2 | 7/2 | 5/2 | 3/2 |
---|---|---|---|---|
5/2, −1 | ||||
3/2, 0 | ||||
1/2, 1 | ||||
−1/2, 2 |
j m1, m2 | 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 = 5/2, j2 = 5/2
j m1, m2 | 5 |
---|---|
5/2, 5/2 |
j m1, m2 | 5 | 4 |
---|---|---|
5/2, 3/2 | ||
3/2, 5/2 |
j m1, m2 | 5 | 4 | 3 |
---|---|---|---|
5/2, 1/2 | |||
3/2, 3/2 | |||
1/2, 5/2 |
j m1, m2 | 5 | 4 | 3 | 2 |
---|---|---|---|---|
5/2, −1/2 | ||||
3/2, 1/2 | ||||
1/2, 3/2 | ||||
−1/2, 5/2 |
j m1, m2 | 5 | 4 | 3 | 2 | 1 |
---|---|---|---|---|---|
5/2, −3/2 | |||||
3/2, −1/2 | |||||
1/2, 1/2 | |||||
−1/2, 3/2 | |||||
−3/2, 5/2 |
j m1, m2 | 5 | 4 | 3 | 2 | 1 | 0 |
---|---|---|---|---|---|---|
5/2, −5/2 | ||||||
3/2, −3/2 | ||||||
1/2, −1/2 | ||||||
−1/2, 1/2 | ||||||
−3/2, 3/2 | ||||||
−5/2, 5/2 |
SU(N) Clebsch–Gordan coefficients
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
- ^ 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.
- ^ 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.
- ^ Mathar, Richard J. (2006-08-14). "SO(3) Clebsch Gordan coefficients" (text). Retrieved 2012-10-15.
- ^ (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.
- ^ Weissbluth, Mitchel (1978). Atoms and molecules. ACADEMIC PRESS. p. 28. ISBN 0-12-744450-5. Table 1.4 resumes the most common.
- ^ 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.
External links
- 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