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Does anybody know why the eigenstates common to the casimir operator $L^2$ and the $z$-axis angular momentum $L_z$ have no $r$ dependence (they are written $\psi_{m,l} (\theta, \phi)$)?

Is it because $L^2$ and $L_z$ only depend on $\theta$ and $\phi$? I have the feeling it is because of this, but I cannot convince myself.

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