9.10: 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
\[ \Delta = \frac{xmax - xmin}{n-1} \nonumber \]
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 xi = 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
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
\[ \psi (k) = eigenvec (H, E_k) \nonumber \]
Display results:
First two even solutions:
First two odd solutions: