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. 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

:$, j_1,j_2;j,m\rangle = \sum_^ \sum_^ , j_1,m_1;j_2,m_2\rangle \langle j_1,j_2;m_1,m_2\mid j_1,j_2;j,m\rangle$

Explicitly:

:
\begin
& \langle j_1,j_2;m_1,m_2\mid j_1,j_2;j,m\rangle \\ pt= & \delta_ \sqrt\ \times \\ pt&\sqrt\ \times \\ pt&\sum_k \frac.
\end

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

For brevity, solutions with and are omitted. They may be calculated using the simple relations

: $\langle j_1,j_2;m_1,m_2\mid j_1,j_2;j,m\rangle=(-1)^\langle j_1,j_2;-m_1,-m_2\mid j_1,j_2;j,-m\rangle.$

and

:$\langle j_1,j_2;m_1,m_2\mid j_1,j_2;j,m\rangle=(-1)^ \langle j_2,j_1;m_2,m_1\mid j_2, j_1;j,m\rangle.$

Specific values

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

When , the Clebsch–Gordan coefficients are given by $\delta_\delta_$.

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.

web interface for tabulating SU(N) Clebsch–Gordan coefficients
is readily available.

References

External links

* Online, Java-base
Clebsch–Gordan Coefficient Calculator
by Paul Stevenson

Other formulae
for Clebsch–Gordan coefficients.
*
Web interface for tabulating SU(N) Clebsch–Gordan coefficients

{{DEFAULTSORT:Table of Clebsch-Gordan coefficients
Representation theory of Lie groups
Clebsch–Gordan coefficients