3 consider a free particle wave packet for which the wave number distr

Question

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!