Imagine we have a system consisting of an electron in an external magnetic field. The electrons magnetic dipole moment will interact with this magnetic field, and the energy of the interaction can be modelled as $$ H = \omega_0 S_z, $$ where $\omega_0 = \frac{eB_0}{2mc}$, just a constant. I have also assumed that the magnetic field is in the $z$ direction, i.e. $\vec{B} = B_0 \hat{k}$, so it only picks out the component of the spin in the $z$ direction.
Thus, it is very easy to find the energy values of this Hamiltonian, since the energy eigenstates are just the same states of the operator $S_z$. We find $$ H |\pm z\rangle = \pm \omega_0 \frac{\hbar}{2}|\pm z\rangle.$$ These are the energy levels for the system. You can note that there is a negative energy level $E_{-} = -\frac{\hbar \omega}{2}$. My question is what is the significance of a negative energy level, because I am not so sure what that means. I understand that negative energies in the presence of a potential (in the S.E.) signify bound states, but here we are not concerned with that. So, what does an energy level mean here, and in general what do negative energies represent physically?