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$\langle z\rangle=\int_0^\infty r^3dr[\int_0^{2\pi}d\phi\int_0^\pi \sin \theta \cos \theta d\theta]=0$ As the $\theta$ integration is zero. But the symmetry argument is clear if the integration is written is Cartesian coordinates.In that case $$\langle z\rangle=\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty z dz |\psi|^2 dx dy$$ As you ...