The discussion revolves around deriving a specific identity related to Clebsch–Gordan coefficients, particularly in the context of angular momentum coupling. Participants are exploring the mathematical formulation and implications of these coefficients.
Participants have not reached a consensus on the identity being derived, and multiple approaches or starting points are suggested without resolution.
The discussion includes complex mathematical expressions and conditions that may not be fully resolved, such as the dependence on the non-negativity of factorial arguments and the specific cases being considered.
Hi, DrClaude.DrClaude said:
The general expression for the CG coefficients is \begin{align*}C_{j_{3}m_{3}}(j_{1}j_{2};m_{1}m_{2}) &= \delta_{m_{3},m_{1}+m_{2}} \left( \frac{(2j_{3}+1) (j_{1} + j_{2} -j_{3})!(j_{1} + j_{3} - j_{2})!(j_{2} + j_{3} - j_{1})!}{(j_{1}+j_{2}+j_{3}+1)!}\right)^{1/2} \\ & \times \left( \prod_{i = 1}^{3}[(j_{i} - m_{i})!] \prod_{i = 1}^{3}[(j_{i} + m_{i})!]\right)^{1/2} \\ & \times \sum_{n} \frac{(-1)^{n}}{n! (j_{1} + j_{2} - j_{3} - n)!(j_{1} - m_{1} - n)!(j_{2} - m_{2} - n)!(j_{3} - j_{1} - m_{2} + n)!(j_{3} - j_{2} + m_{1} + n)!} , \end{align*} where we take the sum over n when non-of the arguments of factorials are negative. The case you are considering: j_{1} = j_{3} \equiv j, \ j_{2} = 1, \ m_{1} = m_{3} \equiv m, \ m_{2} = 0, you sum over n = 0 , 1.victor01 said: