## On conditions for Clebsh-Gordan coefficients to be zero

Posted by peeterjoot on November 23, 2011

# Motivation.

In section 28.2 of the text [1] 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

# The action.

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).

# References

[1] BR Desai. *Quantum mechanics with basic field theory*. Cambridge University Press, 2009.

## Leave a Reply