[UPSC IFoS 2023] Schrodinger’s Wave equation and applications (Q128, 10+10=20M)

(i) Derive an expression for the period of Bloch oscillation for a one-dimensional crystal having lattice period a and electric field \varepsilon. (ii) Consider an electron in a perfectly periodic lattice, wherein the energy-wavenumber relationship in the first Brillouin zone is expressed as E = \frac{\hbar^2 k^2}{5\mathrm{m}_e} where \mathrm{m}_e is the mass of an electron in free space. Find the effective mass of the electron and hence write down the time-independent Schrodinger equation for the electron. Also determine the velocity of the electron. Ignore all interactions except between the electron and the lattice.