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402: Numerical Solutions for a Particle in a V-Shaped Potential Well

  • Page ID
    135870
  • Schrödinger's equation is integrated numerically for a particle in a V-shaped potential well. The integration algorithm is taken from J. C. Hansen, J. Chem. Educ. Software, 8C2, 1996.

    Set parameters:

    n = 100 xmin = -4 xmax = 4 \( \Delta = \frac{xmax - xmin}{n-1}\) \( \mu\) = 1 Vo = 2

    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) \( \Delta\)

    Vi, i = Vo |xi| Ti,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))

    Selected eigenvalues: m = 1 .. 6

    Em =

    \( \begin{array}{|r|}
    \hline \\
    1.284 \\
    \hline \\
    2.946 \\
    \hline \\
    4.093 \\
    \hline \\
    5.153 \\
    \hline \\
    6.089 \\
    \hline\\
    7.030 \\
    \hline
    \end{array} \)

    Display solution:

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

    For V = axn the virial theorem requires the following relationship between the expectation values for kinetic and potential energy: <T> = 0.5n<V>. The calculations below show the virial theorem is satisfied for this potential for which n = 1.

    \( \begin{pmatrix}
    "Kinetic~Energy" & "Potential~Energy" & "Total~Energy" \\
    \psi (1)^{T} T \psi(1) & \psi (1)^{T} V \psi(1) & E_{1} \\
    \psi (2)^{T} T \psi(2) & \psi (2)^{T} V \psi(2) & E_{2} \\
    \psi (3)^{T} T \psi(3) & \psi (3)^{T} V \psi(3) & E_{3}
    \end{pmatrix} = \begin{pmatrix}
    "Kinetic~Energy" & "Potential~Energy" & "Total~Energy" \\
    0.428 & 0.857 & 1.284 \\
    0.982 & 1.964 & 2.946 \\
    1.365 & 2.728 & 4.093
    \end{pmatrix} \)