4. Let (r, y, z) be a complex wavefunction in three dimensions, where (r,y, 2) are the Cartesian coordinates. Define the three-dimensional Fourier transform as o(ks, ky, k=) = (x, y.=) exp(-iksa - ikyy - ik,z)didydz. (12) (a) Find the inverse Fourier transform, i.e., find to(r, y, 2) in terms of o(kx, ky, k=). (1 mark) (b) In the vector notation, the Fourier transform can also be written as (k) = (r)exp (-ik . r)dr, (13) where r = ci + yy + 22 and k = kra + kyy + k,2. Find the Fourier transform of the Laplacian V'd(r) = V . Vu(r) = 02 + Or2 a2 2 $5(r) (14) in terms of o(k ) and k. (1 mark) (c) Consider the Schrodinger equation for a free particle: ih- Orb(r, t) h2 at V'i ( r , t ) . 2m (15) Find the differential equation for o ( k, t ) = (r,t) exp(-ik . r)d'r. (16) (1 mark) (d) Solve for o(k, t) in terms of the initial condition o(k, 0). (1 mark) (e) Solve for th(r, t) in terms of the initial condition th(r, 0). (2 marks)