# 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 x_{i} = xmin + (i - 1) \( \Delta\)

V_{i, i} = V_{o} |x_{i}| 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))

Selected eigenvalues: m = 1 .. 6

E_{m} =

\( \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 = ax^{n} 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} \)