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 [cite journal |last=Baird |first=C.E. |coauthors=L. C. Biedenharn |title=On the Representations of the Semisimple Lie Groups. III. The Explicit Conjugation Operation for SU_n |journal=J. Math. Phys. |volume=5 |year=1964 |month=October |pages=1723–1730 |doi=10.1063/1.1704095 |url=http://link.aip.org/link/?JMAPAQ/5/1723/1 |accessdate=2007-12-20] . Tables with the same sign convention may be found in the Particle Data Group's "Review of Particle Properties" [cite journal |last=Hagiwara |first=K. |coauthors="et al." |title=Review of Particle Properties |journal=Phys. Rev. D |volume=66 |year=2002 |month=July |pages=010001 |doi=10.1103/PhysRevD.66.010001 |url=http://pdg.lbl.gov/2002/clebrpp.pdf |format=PDF |accessdate=2007-12-20] and in online tables. [cite web |last=Mathar |first=Richard J. |title=SO(3) Clebsch Gordan coefficients |date=2006-08-14 |url=http://www.strw.leidenuniv.nl/~mathar/progs/CGord |format=text |accessdate=2007-12-20] .

Formulation

The Clebsch-Gordan coefficients are the solutions to

:$langle\; j\_1j\_2;m\_1m\_2|j\_1j\_2;jm\; angle=?$,explicitly

$delta\_\{m,m\_1+m\_2\}sqrt\{frac\{(2j+1)(j+j\_1-j\_2)!(j-j\_1+j\_2)!(j\_1+j\_2-j)!\}\{(j\_1+j\_2+j+1)!\; imes$

$sqrt\{(j+m)!(j-m)!(j\_1-m\_1)!(j\_1+m\_1)!(j\_2-m\_2)!(j\_2+m\_2)!\}\; imes$

$sum\_k\; frac\{(-1)^k\}\{k!(j\_1+j\_2-j-k)!(j\_1-m\_1-k)!(j\_2+m\_2-k)!(j-j\_2+m\_1+k)!(j-j\_1-m\_2+k)!\}.$

The summation is extended over all integral "k" for which the argument of every factorial is nonnegative. [(2.41), p. 172, "Quantum Mechanics: Foundations and Applications", Arno Bohm, M. Loewe, New York: Springer-Verlag, 3rd ed., 1993, ISBN 0387953302.]

For brevity, solutions with m < 0 are omitted. They may be calculated using the relation

:$langle\; j\_1j\_2;m\_1m\_2|j\_1j\_2;jm\; angle=(-1)^\{j-j\_1-j\_2\}langle\; j\_1j\_2;-m\_1,-m\_2|j\_1j\_2;j,-m\; angle$ .

=j₁=1/2, j₂=1/2=

=j₁=1, j₂=1/2=

=j₁=1, j₂=1=

=j₁=3/2, j₂=1/2=

=j₁=3/2, j₂=1=

=j₁=3/2, j₂=3/2=

=j₁=2, j₂=1/2=

=j₁=2, j₂=1=

=j₁=2, j₂=3/2=

=j₁=2, j₂=2=

=j₁=5/2, j₂=1/2=

=j₁=5/2, j₂=1=

=j₁=5/2, j₂=3/2=

=j₁=5/2, j₂=2=

References

External links

* Online, Java-based [http://www.gleet.org.uk/cleb/cgjava.html Clebsch-Gordan Coefficient Calculator] by Paul Stevenson
* [http://functions.wolfram.com/HypergeometricFunctions/ClebschGordan/06/01/ Other formulae] for Clebsch-Gordon coefficients.