Recursion relation for C-G coefficients

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The discussion focuses on the Clebsch-Gordan coefficients and their properties as outlined in J.J. Sakurai's "Modern Quantum Mechanics." It raises questions about why terms in the recursion relation do not vanish when m1 + m2 equals m + 1 or m - 1, despite the coefficients being zero unless m equals m1 + m2. Additionally, it queries why, in a specific example, m1 + m2 equals m. The confusion stems from the application of the recursion relation and the conditions under which the coefficients are defined. Clarification on these points is sought to better understand the underlying quantum mechanics concepts.
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From J.J. Zakurai page no.222-225:

We know that Clebsch-Gordan coefficients vanish unless m=m1+m2. Then,

(1) why is that the terms in recursion relation doesn't vanish? since m1+m2=m+1 or m1+m2=m-1 in eqn.(3.7.49).

(2) why is again in the example shown in eqn.(3.7.54), m1+m2=m?

I couldn't understand the whole thing. Somebody please help.
 
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Excuse me, the book is- "Modern Quantum Mechanics" by J.J. Sakurai
 
what edition?
 
It is the 'revised edition' printed in 2005.
 

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