Use your favorite computer program (e.g. Igor, Matlab, Mathematica) to solve the 1D Schrodinger equation by finite differences. Hint: you'll set up an N N matrix, where the elements index the position x, and N is the mesh size. Diagonalize the matrix to find the eigenvalues and eigenvectors. The eigenvalues correspond to the energies and the eigenvectors correspond to the wavefunction (z). Print out your code where you assemble and diagonalize the matrix. Then run your calculation for the following conditions (a) Check that you have set everything up correctly by calculating the lowest three energies of the harmonic mw212, where m is the mass and the frequency. For simplicity, set all the m =-1). You should have energies En-nw (n + )-(n + ) oscillator, ie. U(x) coefficients to one (h (b) Plot the potential and Vn(x)2 for the first three wavefunctions n 1,2,3. Recall that ( is the probability of finding the particle (electron) at position . On your plot, offset the wavefunctions by their corresponding energies E (c) We can use this to approximate the electronic structure of a material. The Kronig-Penney model approximates the atomic potential as a lattice of square wells. Normally Kronig-Penney is done for an infinite crystal, but here you are set up for finite size .e. a slab or a surface). Try a few reasonable values of parameters (atomic spacings (a + b) of order 3 , well height of order 100 eV) and print plots of U(x) andfor the bound states. How does the energy bandwidth changes from the lowest energy states to the highest ones, and conment on what this means for a real material? Think core levels versus valence bands 7 0