513.001 Molecular and Solid State Physics

Hamiltonian matrix

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

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.

The expectation value of the energy is,

Where

Minimize the expectation value of the energy with respect to c₁*, c₂*, and c₃*. The derivatives are:

Setting these derivatives equal to zero results in the following matrix equation.

Use a computer algebra program to solve this problem this matrix equation. Write down the approximate ground state and first two excited states of the perturbed potential and give their eigen energies.

If you want to find the best solution of a Hamiltonian in terms of some orthonormal set of eigen functions, you can immediately construct the corresponding Hamiltonian matrix. The best solution will be given by the eigen vector of the matrix with the lowest eigen value.