# 404: Numerical Solutions for a Double-Minimum Potential Well

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

$\Delta = \frac{xmax - xmin}{n-1}$

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 xi = xmin + (i - 1) $$\Delta$$

$V_{i,~j} = if \bigg[ i =j,~ b(x_i)^4 - c(x_i)^2 ,~0 \bigg]$

$T_{i,~j} = if \bigg[ i=j, \frac{ \pi ^{2}}{6 \mu \Delta ^{2}}, \frac{ (-1)^{i-j}}{ (i-j)^{2} \mu \Delta^{2}} \bigg]$

Hamiltonian matrix:

$H = T + V$

Calculate eigenvalues: E = sort(eigenvals(H))

Display three eigenvalues: m = 1 .. 5

Em =

$$\begin{array}{|r|} \hline \\ -6.64272702 \\ \hline \\ -6.64062824 \\ \hline \\ -2.45118605 \\ \hline \\ -2.3155705 \\ \hline \\ 0.41561275 \\ \hline \end{array}$$

Calculate selected eigenvectors:

k = 1 .. 4

$\psi (k) = eigenvec (H, E_k)$

Display results:

First two even solutions:

First two odd solutions: