Answer Happy

please solve problem c
packets. Consider a free particle of mass m in one spatial dimension. The Hamiltonian is (1) 2m dr² a.) Let us consider the initial wave function in a form of a Gaussian wave packet centred around zo with a width 06: (1-10)² 207 (2) Calculate the time evolution of this wave function e(t.r) for a free particle with the initial condition (0,r) = o(r) given by Eq.(2). From this calculation show that le(t. 2)² = N(t)exp (- (2-1)²) (3) and obtain expressions for N(t) and (t). b.) In the previous part you dealt with a wave packet whose center does not move. Now we consider a moving wave packet. The initial condition in this case is (0,2)=√√0 exp (iPoz/h) exp(- (220)²) Show that the resulting time evolution of this wave function is such that le(t, x)² = N(t) exp(-(x-ze(0))²), (5) where N(t) and (t) are the same as before, but Ic(t) = 1o+ Pot/m. In other words, the center of this wave packet is moving with the velocity to Po/m. e.) Now we return to the applicability limits of the classical Drude model. The classical approximation works as long one can replace colliding classical particles by colliding quantum wave packets. The limitation here comes from the fact that quantum wave packets spread between the collisions according to the above result for r(t). The approximation breaks down, when o(t) grows faster than the center of the wave packet moves, which, in turn, imposes a restriction on the minimum value of do. On the other hand, 70 must be smaller than the mean free path. Find the minimum mean free path for the applicability of the Drude model to a charge carrier having mass m= 10-27 g in two cases: (i) when the charge carrier moves with thermal velocity at temperature T=300K (the case of semiconductors), or (ii) when the charge carrier moves with the Fermi-velocity corresponding to the Fermi-energy ErleV (one electronvolt). 1 Hint: While solving this problem, you will often need the following integral: Le Bdr = √FA (6) Note that it is true not only for real, but for complex values of A as well.