Exercise 2.1: Infinite potential well

$\bullet$ Write a computer program in which you fill the overlap and Hamiltonian matrices for this problem. Use standard software to solve the generalized eigenvalue problem. Notice that the matrices are Hermitian, so only the upper, or lower triangular parts have to be calculated (including the diagonal).

$\bullet$ Compare results with exact analytic solutions. These are given by

$$\psi_n(x) = \left\{ \begin{array}{ccc} \cos{(k_nx)} & n & {\... ... n & {\mathrm even and positive} \end{array}\right.$$ (34)

with $k_n=n\pi/2, n=1,2...$, and the corresponding energies are given by

$$E_n = k_n^2=\frac{n^2\pi^2}{4}$$ (35)

For each eigenvector ${\bf c}$, the function $\sum_{p=1}^{N} = c_p\phi_p(x)$ should approximate an exact eigenfunction. They can be compared by displaying both graphically. Carry out the comparison for various numbers of basis states kept.