3) Consider a free particle wave packet for which the wave number distribution A(K) = { C for - Ak ≤ k ≤ Ak 0 elsewhere with constant C > 0.

a) Determine the constant C so that A² (k) is normalized. b) Sketch A(k). c) Determine the associated spatial probability density (x, 0)|² at time t = 0. Determine the value of y(x,0)|² at x = 0. Use these results to sketch |(x,0)|². d) Approximating the width of A²(k) as Ak and the width Ax as the distance from the origin to the first minimum of y(x, 0)|², determine the product Ax Ap at time t = 0 and comment on your result. e) Write down an explicit integral expression for (x, t). You do not need to solve this integral!