# 408: Numerical Solutions for the Double Morse Potential

- Page ID
- 137732

Schrodinger's equation is integrated numerically for the first four energy states for the double Morse oscillator. The integration algorithm is taken from J. C. Hansen, ** J. Chem. Educ. Software, 8C2**, 1996.

Set parameters:

n = 200

xmin = -10

xmax = 10

\[ \Delta = \frac{xmax - xmin}{n-1}\]

\( \mu\) = 1

D = 2

\( \beta\) = 1

x_{0} = 1

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) \( \Delta\)

\[ V_{i,~j} = if \bigg[ i =j,~ D \big[ 1 - exp \big[ - \beta (|x_i| - x_0) \big] \big] ^2 ,~0 \bigg] \]

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

Hamiltonian matrix: H = T + V

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

Display four eigenvalues: m = 1 .. 4

E_{m} =

\( \begin{array}{|r|}

\hline \\

0.8092 \\

\hline \\

0.9127 \\

\hline \\

1.8284 \\

\hline \\

1.8975 \\

\hline

\end{array} \)

Calculate associated eigenfunctions:

k = 1 .. 4

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

Plot the potential energy and bound state eigenfunctions:

\[ Vpot_i = V_{i,~i}\]