Let $H$ be the Hamiltonian of a specific atom (not hydrogen) and $J$ the total angular momentum. Since $H$ and $J$ commute, they have common eigenstate. So we can label the atomic states by their energy and total angular momentum $\phi_E^J$.

My question is, suppose we have a state $\phi_E^J$ and another state $\phi_{E'}^{J'}$, do we have necessarily that $E\neq E'$, that is, do states with different total angular momentum have necessarily different energies?


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