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As we know, if we solve the schrodinger equation, the ground state wavefunction is independent of theta and psi. We find the expectation value of ground state orbital angular momentum is zero. But if we don't do any mathematical calculation, can we conlude that?

For example, Due to symmetry of coulomb potential, Hydrogen atom (one proton plus one electron) wavefunction must be symmetric or antisymmetric (for ground state, there is no degeneracy). Then I am stuck. How can I conclude that only by virtue of physical concept or symmetry? Does symmetry means spherical symmetry?

thanks

xf