Consider a particle in a 1D periodic lattice. The lattice constant is \( a \) and the hopping frequency is \( c \) . There is an applied electric field \( f \) .

(1) Write the Hamiltonian \( H(p,x) \) with \( f \) that is derived from a scalar potential \( V(x) \) .

(2) Derive the classical equations of motion and solve them. What is the period of the oscillation? What is the spatial amplitude of the oscillation?

(3) Write the Hamiltonian \( \tilde{H}(p;t) \) with \( f \) that is derived from a vector potential \( A(t) \) . What is the gauge transformation of the basis that relates the two Hamiltonians?

(4) Find an explicit expression in terms of Bessel function for the evolving probability distribution \( \rho(x;t) \)

(5) What is the solution \( \rho(x;t) \) in the zero field limit? Plot or describe in words how the spreading profile look like in this case.

(6) Verify that for finite field the amplitude of the Bloch oscillations is in agreement with the classical result.