Consider a mass \( m \) particle in a 1D periodic lattice with structure \( (ab)^{\#} \) where \( a \) and \( b \) are the lengths of the bonds. What are the hopping amplitudes \( c_a \) and \( c_b \) between the sites. Write the Hamiltonian matrix \( H \) assuming that all sites have the same binding energy. Define the displacement operator \( D \) with which \( H \) commutes. Find the two eigenstates of \( H \) that have \( e^{i\phi} \) periodicity. Tip: the requested wavefuntion is fully determined by two amplitudes \( (\psi_1,\psi_2) \) for which you should write a reduced eigen-equation. Use the same procedure find the eigenvalues of a lattice that has the structure \( (aab)^{\#} \) Explain in what sense the reduced eigen-equation describes a ring.