START FROM (1-2) by using THE MATLAB PLEASE

1-1. (20 pts) Construct a N by N Hamiltonian matrix by applying the following conditions, in a similar way that we did in class. For this problem, you are not required to run the simulation. Just write down your answer (Hamiltonian matrix). Show your work to get the full credit. - Select a one-dimensional discrete lattice with N points spaced by a - Apply the boundary condition of (x = 0) = (x = N+1) = 0 - Set the potential energy U(x) = 0 for 0 < x < N+1

1-2. (60 pts) Use any available simulation tool like MATLAB to find the eigenvalues (i.e., energies) and the corresponding eigenvectors (i.e., wavefunctions) 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 or sampling) = 100 and a (mesh spacing) = 1e10 (m) for your simulation. Follow the steps below from (a) to (e) to solve this problem.

(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_bar2 2n 2 /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)

1-3. (20 pts) Re-write your Hamiltonian (dont need to run the simulation. Just write your answer) when (a) the periodic boundary condition is applied, i.e., (x = 0) = (x = N) and (x = 1) = (x = N+1). (10 pts) (b) We have so far assumed the potential energy is zero. What about the potential energy U(x) is not zero, i.e. U (x) = U0 for 0