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Chemistry LibreTexts

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 V0 = 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 xi = 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

    Em =

    \( \begin{array}{|r|}
    \hline \\
    0.63423174 \\
    \hline \\
    2.39691438 \\
    \hline \\
    4.4105828 \\
    \hline
    \end{array} \)

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

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

    Screen Shot 2019-02-06 at 1.25.31 PM.png