513.001 Molecular and Solid State Physics

Return to
problem list

Wolfram Alpha

Physical
Constants

Periodic System
of Elements

      

Hamiltonian matrix

The infinite square well problem has the following eigen function solutions and eigen energies.

$\large -\frac{\hbar^2}{2m}\nabla^2\psi (x) = E\psi (x)$

$\large \psi_n = \sqrt{\frac{2}{L}}\sin \left(\frac{\pi nx}{L}\right)\hspace{1.5cm}n=1,2,3,\cdots$

$\large E_n=\frac{\pi^2\hbar^2 n^2}{2mL^2}$

Consider the perturbed potential

We seek the best solution to the this perturbed potential in terms of a linear superposition of the first three infinite square well eigen states.

$\large \psi = c_1\psi_1+c_2\psi_2 + c_3\psi_3$

Construct the Hamiltonian matrix and solve the resulting eigenvalue equation to find the eigenvalues of and eigenvectors.