9.10: Numerical Solutions for a Double-Minimum Potential Well

Last updated
Save as PDF

Page ID: 137728

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

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:

Screen Shot 2019-02-06 at 7.32.57 PM.png

First two odd solutions:

Screen Shot 2019-02-18 at 12.23.10 PM.png