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Let $H$ be the Hamiltonian of a specific atom 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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    $\begingroup$ You might have heard of hydrogen atom... $\endgroup$ Nov 11 '20 at 16:18
  • $\begingroup$ I mean atoms different from hydrogen $\endgroup$ Nov 11 '20 at 16:19
  • $\begingroup$ Given a fixed momentum of inertia, the rotational kinetic energy of a system is monotonically related to the the magnitude of the angular momentum. Therefore, if the system has different angular momentum, it has different energy. $\endgroup$
    – Zhe
    Nov 11 '20 at 17:59
  • $\begingroup$ The magnitude of the angular momentum is $J=\hbar\sqrt{j(j+1}$ and energy is momentum squared divided by twice the mass (or moment of inertia for rotation) then energy is $J^2/(2I) $ or $\hbar^2 j(j+1)/2I$ so depends on $J$, so different $J$ different energy. This is just the expression for particle on a sphere or rigid rotor. $\endgroup$
    – porphyrin
    Nov 12 '20 at 9:14
  • $\begingroup$ well we could have accidentally symmetry $\endgroup$ Nov 12 '20 at 9:29

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