On conditions for Clebsh-Gordan coefficients to be zero
Posted by peeterjoot on November 23, 2011
In section 28.2 of the text  is a statement that the Clebsh-Gordan coefficient
unless . It appeared that it was related to the operation of , but how exactly wasn’t obvious to me. In tutorial today we hashed through this. Here’s the details lying behind this statement
Recap on notation.
We are taking an arbitrary two particle ket and decomposing it utilizing an insertion of a complete set of states
with and fixed, this is written with the shorthand
so that we write
We have two ways that we can apply the operator to . One is using the sum above, for which we find
We can also act directly on and then insert a complete set of states
This provides us with the identity
This equality must be valid for any , and since all the kets are linearly independent, we must have for any
We have two ways to get this zero. One of them is a condition, and the other is for the CG coeff to be zero whenever .
It’s not a difficult argument, but one that wasn’t clear from a read of the text (at least to me).
 BR Desai. Quantum mechanics with basic field theory. Cambridge University Press, 2009.