# 399: Particle in a Finite Potential Well

- Page ID
- 135867

## Numerical Solutions for the Finite Potential Well

Schrödinger's equation is integrated numerically for the first three energy states for a finite potential well. The integration algorithm is taken from J. C. Hansen, J. Chem. Educ. Software, 8C2, 1996.

Set parameters:

n = 100 | xmin = -3 | xmax = 3 | \( \Delta = \frac{ xmax - xmin}{n-1}\) |

\( \mu\) = 1 | lb = -1 | rb = 1 | V_{0} = 4 |

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) \( \Delta\) |

\( V_{i,i} = if[ (x_{i} \geq lb) (x_{i} \leq rb), 0, V_{0}]\) \( T_{i, j} = if [i=j, \frac{ \pi^{2}}{6 \mu \Delta^{2}}, \frac{(-1)^{i-j}}{(i-j)^{2} \mu \Delta^{2}} ]\)

Form Hamiltonian energy matrix: H = TV

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

Display three eigenvalues: m = 1 .. 3

E_{m} =

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

\hline \\

0.63423174 \\

\hline \\

2.39691438 \\

\hline \\

4.4105828 \\

\hline

\end{array} \)

Calculate associated eigenfunctions: k = 1 .. 2 \( \psi\)(k) = eigenvec(H, E_{k})

Plot the potential energy and bound state eigenfunctions: \( V_{pot1} := V_{i,i}\)