# Expectation value of coordinate mixed operator with ground state

I have a Hamiltonian of the Hydrogen atom: $$H=H_0+H_1+H_2$$ , when: $$H_0$$ is the hamiltonian from central force and from electron momentum , $$H_1$$ is the relativistic kinetic fixing, and $$H_2$$ is the Darwin element.

We are in the ground state, and it given to us that this state have degeneracy of 2 so in $$|n,l,m_l,m_s>$$ base, we are in the state of $$|1,0,0,\pm 1/2 >$$

in this state I need to find the expectation value of the operator: $$A=XYS_xS_y+XZS_xS_z+YZS_yS_z$$ when :$$S_i$$ represent spin in $$i$$ coordinate

$$<1,0,0,\pm 1/2|A|1,0,0,\pm 1/2>=?$$

I think it's need to be zero , but I can`t proof why? it's somehow related with the fact that the state have some symmetry in the angular momentum $$l$$. I don't get it...