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Chemistry LibreTexts

404: 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

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

    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) \( \Delta\)

    \[ V_{i,~j} = if \bigg[ i =j,~ b(x_i)^4 - c(x_i)^2 ,~0 \bigg] \]

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

    Hamiltonian matrix:

    \[H = T + V\]

    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

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

    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