Sakurai proof Wigner-Eckart theorem

[detste]

On page 240 in Sakurai's book "Modern Quantum Mechanics" (Revised edition) you can find a proof of the Wigner-Eckart theorem. I don't understand how you can deduce equation (3.10.36) from equation (3.10.35). I also don't understand why this proportionality factor is independent of m, q and m'. Can someone clarify this?

 

[Jolb]

Well first of all, 3.10.36 follows from 3.10.35 because the recursion relation for the T_q^(k) matrix elements is the same as the recursion relation for the Clebsch-Gordon coefficients.

The recursion relation for the CG coefficients gives us the ratio between any two CG coefficients. But there is nothing which dictates their absolute magnitudes (aside from an arbitrary convention). If the T_q^(k) matrix elements are known to follow the same recursion relation, then we know the ratios between all of them, too, and these ratios are exactly the same as those between the CG coefficients. However, the absolute magnitude of these matrix elements is not dictated by the recursion relation, and the absolute magnitude need not be equal to that of the CG coefficient. Thus the only difference that can exist between the T_q^(k) matrix elements and the CG coefficients is an overall scale factor (by overall I mean the same scale factor relates all T_q^(k) matrix elements with their associated CG coefficient.)

Now since the CG coefficient recursion relation dictates all of the CG coefficients (not just for a given m, q, or m'), then this overall scale factor must not depend on m, q, or m'.