In class, we derived the rather remarkable results that if we know the bandstructure E(k) of any crystal, the velocity of an electron in state k is its group velocity vy(k) = %%l with suitable generalization to higher dimensions, and the motion of the states in kspace is given simply by F' = % where F' is the force. In this problem, consider an electron on a ring of length L moving through an extremely weak periodic crystal po- tential of lattice constant a such that the periodicity of the crystal is imprinted on the electron wavefunctions. (a) Show that in the limit of a vanishing crystal potential, the wavefunction of the electron must be of the form (z) = e'\"a)m, where G = %'rn is a reciprocal lattice vector with n = ... 1,0, +1,.... (b) The Bloch wavefunction is of the form 1 (z) = e**u(z) such that u(z + a) = u(x). Show that the electron wavefunction of part (a) is in the Bloch form. Identify u(z) and prove that indeed u(z + a) = u(z). (c) Show that the allowed electron eigenvalues are E(k) = W. Write expressions for several values of G, and sketch the corresponding nearly free electron energy bands for 3%