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

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:

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.

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