Understanding Angular Momentum States and Clebsh-Gordan Coefficients

[spookyfish]

When we add the angular momenta of two particles, J1 and J2, we get that the resulting total angular momenta is in the range
|J1-J2| < J < J1+J2

but according to the Clebsh-Gordan table some coefficients are zero. Does it mean that not all combinations between |J1-J2| and J1+J2 are possible?

 

[dauto]

No, they are all possible. The Clebsh-Gordan table also takes the components (the m's) into consideration.

 

[spookyfish]

ok, but for a given m - some of them might not be possible

 

[Bill_K]

It's due to symmetry. For example, rotate through 180 degrees, turning z → -z. This reverses the sign of each m value. There's an identity

(j1 -m1 j2 -m2|j3 -m3) = (-1)^{j1 + j2 + j3} (j1 m1 j2 m2|j3 m3)

So in particular for m1 = m2 = m3 = 0 the Clebsch-Gordan coeffiicent will vanish if j1 + j2 + j3 is odd.

 

[dauto]

ok, but for a given m - some of them might not be possible

That's correct