In the case of the hydrogen atom (with the assumption that there is no spin-orbit coupling), the observables that commute with Hamiltonian are the orbital angular momentum, spin angular momentum, the sum of the spin angular momentum and orbital angular momentum, and the components of the above angular momenta.
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It implies that a physical quantity is conserved if its Poisson Bracket with the Hamiltonian is zero and it does not depend on time explicitly.
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If the spin orbit interaction is taken into account, we have to add an extra term in Hamiltonian which represents the magnetic dipole interaction energy.
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Systems which can be labelled by good quantum numbers are actually eigenstates of the Hamiltonian.
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If such operators commute with the Hamiltonian, then their expectation value remains constant with time.
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