February 2, 1998

PHY 742 – Notes on Clebsch-Gordan coefficients

Reference: Abramowitz & Stegum – pg. 1006 Original formula:

< j_{1}j_{2}m_{1}m_{2}|j_{1}j_{2}jm >= δ_{m,m_{1}+m_{2}} \sqrt{\frac{(j_{1}+j_{2}−j)!(j_{1}−j_{2}+j)!(−j_{1}+j_{2}+j)!(2j+1)}{(j_{1}+j_{2}+j+1)!}}  (1)

× ∑_{k} ^{\frac{(−1)^{k}\sqrt{(j_{1}+m_{1})!(j_{1}−m_{1})!(j_{2}+m_{2})!(j_{2}−m_{2})!(j+m)!(j−m)!}}{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)!}}

Working formula:

< j_{1}j_{2}m_{1}m_{2}|j_{1}j_{2}jm >= δ_{m,m_{1}+m_{2}} \sqrt{\frac{(j_{1}−j_{2}+j)!(−j_{1}+j_{2}+j)!(2j+1)}{(j_{1}+j_{2}−j)!(j_{1}+j_{2}+j+1)!}}  (2)
×\sqrt{\frac{(j_{1}+m1)!(j_{2}−m_{2})!(j+m)!}{(j_{1}−m1)!(j_{2}+m_{2})!(j−m)!}}

× ∑_{k} \frac{(−1)^{k}}{k!} \frac{(j_{1}+j_{2}−j)!}{(j_{1}+j_{2}−j−k)!} \frac{(j_{1}−m_{1})!}{(j_{1}−m_{1}−k)!} \frac{(j_{2}+m_{2})!}{(j_{2}+m_{2}−k)!} \frac{(j−m)!}{(j−j_{2}+m_{1}+k)!(j−j_{1}−m_{2}+k)!}