1-2. (60 pts) Use any available mathematical package like MATLAB to find the eigenvalues and the corresponding eigenvectors for the electron in a box problem that you just modeled in 1-1. Because your Hamiltonian is a N by N matrix, you should get N pairs of eigenvalues and eigenvectors. Take N (# of mesh) = 100 and a (mesh spacing) = 1e-10 (m) for your simulation. (a) Plot the numerically obtained eigenvalues (in eV) as a function of eigenvalue number n. n is the integer number that ranges from 1 to N. For example, if you obtain N number of eigenvalues, n = 1 indicates the first set of eigenvalue and eigenfunction, n = 2 means the second eigenvalue and eigenfunction, etc. (20 pts) (b) Plot the analytical eigenvalues (En as a function of n), using the equation, En = h_bar^{2 2} n² /2mL² where L = (N+1)a, together with the result obtained in (a). In other words, overlap your answer in (b) with that in (a). h_bar is the reduced Planck constant. (5 pts) (c) Describe how well your numerical result in (a) matches the analytical result in (b). (5 pts) (d) Plot 2 (squared eigenvector) for eigenvalue numbers n = 1, 2, 10, and 50. (20 pts) (e) Use the result of (d) to explain any observation you made in (c). (10 pts)