Numerical Solutions for the Finite Potential Well Schrödinger's equation is integrated numerically for the first three energy states for a finite potential well. The integration algorithm is taken from J. C. Hansen, J. Chem. Educ. Software, 8C2, 1996. Set parameters: 0 1 2 3 n = 100 xmin = -3 xmax = 3 \( \Delta = \frac{ xmax - xmin}{n-1}\) \( \mu\) = 1 lb = -1 rb = 1 V0 = 4 Calculate position vector, the potential energy matrix, and the kinetic energy matrix. Then combine them into a total energy matrix. 0 1 2 i = 1 .. n j = 1 .. n xi = xmin + (i - 1) \( \Delta\) \( V_{i,i} = if[ (x_{i} \geq lb) (x_{i} \leq rb), 0, V_{0}]\) \( T_{i, j} = if [i=j, \frac{ \pi^{2}}{6 \mu \Delta^{2}}, \frac{(-1)^{i-j}}{(i-j)^{2} \mu \Delta^{2}} ]\) Form Hamiltonian energy matrix: H = TV Find eigenvalues: E = sort(eigenvals(H)) Display three eigenvalues: m = 1 .. 3 E m = \( \begin{array}{|r|} \hline \\ 0.63423174 \\ \hline \\ 2.39691438 \\ \hline \\ 4.4105828 \\ \hline \end{array} \) Calculate associated eigenfunctions: k = 1 .. 2 \( \psi\)(k) = eigenvec(H, E k ) Plot the potential energy and bound state eigenfunctions: \( V_{pot1} := V_{i,i}\)