# Do different eigenstates of total angular momentum have necessarily different energies?

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?

• You might have heard of hydrogen atom... Nov 11 '20 at 16:18
• I mean atoms different from hydrogen Nov 11 '20 at 16:19
• 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.
– Zhe
Nov 11 '20 at 17:59
• 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. Nov 12 '20 at 9:14
• well we could have accidentally symmetry Nov 12 '20 at 9:29