Table of Clebsch–Gordan coefficients

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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

.

A complete list [5][edit]

 j2=0[edit]

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

 j1=1/2, j2=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=3/2, j2=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=3/2, j2=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=3/2, j2=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=5/2, j2=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=5/2, j2=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=5/2, j2=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=5/2, j2=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

SU(N) Clebsch–Gordan coefficients[edit]

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[edit]

  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:1009.0437Freely accessible. Bibcode:2011JMP....52b3507A. doi:10.1063/1.3521562. Retrieved 2011-04-13. 

External links[edit]