For a free particle in lineland, a momentum eigenstate is also an energy eigenstate: Hlp ) = Elp); Ep = 2m (3) Let us specify a state (at time equal to zero) by its wave function in the momentum basis: y (p, 0) = (ply(t =0)) = Ne-(P-Po)-/46p; No= (2no2)-1/4, (4) with o a real-valued constant. (2a) For the time-evolved state | yr(t) ), evaluate Pave(t) and Ap(t).2 (2b) Find w(x, t) by Fourier transforming the time-evolved wave function in the momentum basis.' Express your final answer in terms of op, h, a time constant . = mh/(20) and the group velocity vg = (dE/dp) = Po/m. In particular, your final expression should depend on x only in the combinations x - vet and x - Ugt/2. (2c) Determine X peak (t), with Xpeak (t) being the position of the peak of ly (x, t) |2. (2d) Determine X ave (t) and Ax(t).4 (2e) Determine Ax(r)/Ax(0) and explain (in one or two sentences) why the wave packet is broadening, bearing in mind that each complex wave in a wave packet is moving with a phase velocity phase given by ei(px-E,t)/h = eip(x-upset)/h. uphase = _ Ep (5) P (2f) Evaluate the product Ax(t)Ap(t) and identify all times for which | yr(t) ) satisfies the lower bound of Heisenberg's uncertainty principle