The goal is to find the eigenstates and the eigenenergies of any 1D Hamiltonian, $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x).$$ To do this, one needs to be able to represent the Hamiltonian operator as a matrix for a given basis. A variety of basis can be chosen. The more physically relevant the basis, the more efficient the computations. However as long as the basis can be systematically increased, one should be able to find the correct eigenstates. This procedure is called “convergence.”