Schrödinger's equation is integrated numerically for a double minimum potential well: \(V = bx^4 - cx^2\). The integration algorithm is taken from J. C. Hansen, J. Chem. Educ. Software, 8C2 , 1996. Set parameters: Increments: n = 100 Integration limits: xmin = -4 xmax = 4 \[ \Delta = \frac{xmax - xmin}{n-1} \nonumber \] Effective mass: \( \mu\) = 1 Constants: b = 1 c = 6 Calculate position vector, the potential energy matrix, and the kinetic energy matrix. Then combine them into a total energy matrix. i = 1 .. n j = 1 .. n x i = xmin + (i - 1) \( \Delta\) \[ V_{i,~j} = if \bigg[ i =j,~ b(x_i)^4 - c(x_i)^2 ,~0 \bigg] \nonumber \] \[ T_{i,~j} = if \bigg[ i=j, \frac{ \pi ^{2}}{6 \mu \Delta ^{2}}, \frac{ (-1)^{i-j}}{ (i-j)^{2} \mu \Delta^{2}} \bigg] \nonumber \] Hamiltonian matrix: \[H = T + V \nonumber \] Calculate eigenvalues: E = sort(eigenvals(H)) Display three eigenvalues: m = 1 .. 5 E m = \( \begin{array}{|r|} \hline \\ -6.64272702 \\ \hline \\ -6.64062824 \\ \hline \\ -2.45118605 \\ \hline \\ -2.3155705 \\ \hline \\ 0.41561275 \\ \hline \end{array} \) Calculate selected eigenvectors: k = 1 .. 4 \[ \psi (k) = eigenvec (H, E_k) \nonumber \] Display results: First two even solutions: First two odd solutions: