Exercise 8.1: Single-slit diffraction

Young's single-slit experiment consists of a wave passing though a small slit, which causes the emerging wavelets to intefere with eachother forming a diffraction pattern. In quantum mechanics, where particles are represented by probabilities, and probabilities by wave packets, it means that the same phenomenon should occur when a particle (electron, neutron) passes though a small slit. Consider a wave packet of initial width 3 incident on a slit of width 5, and plot the probability density $\vert\psi ^2\vert$ as the packet crosses the slit. Generalize the time-evolution equation (88), and the wave packet (80) for 2 dimensions (you need to use two coordinates $x$ and $y$ and the momentum will also have two components $k_x$ and $k_y$). Model the slit with a potential wall:

$$V(x,y)=100 \mathrm{for} x=10,\vert y\vert\geq 2.5.$$