 Solution

At each wavevector the orbital-basis Hamiltonian isA phase choice makes the momentum-independent mixing real and nonnegative. The characteristic equation issoFor $\alpha >0$, an eigenvector obeys $c_{\mathrm{fl}}/c_p=(E_{\pm }-E_p)/\alpha$. At the bare crossing, the normalized states are $(|p⟩\pm |\mathrm{fl}⟩)/\sqrt{2}$ with energies $E_{\mathrm{fl}}\pm \alpha$. The lower state's parabolic-orbital weight iswhich tends to one below the crossing and zero above it. This makes the exchange of band character in the sketches explicit.