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

- Page ID
- 137728

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 x_{i} = 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

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

\[ \psi (k) = eigenvec (H, E_k)\]

Display results:

First two even solutions:

First two odd solutions: