For an angular momentum eigenstate $|l,m\rangle$ Sakurai says that

$$ |l,m\rangle = (-1)^l |l,m \rangle $$

I have a problem in which I have $ J = L + S $ and I want to apply the previous formula. I am dealing with a spin 1/2 particle so I have states like $|\frac{1}{2},\frac{1}{2}\rangle$ (coupled base state). If I use the formula such a state will have an imaginary parity... What is wrong?



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