The importance of basis states: an example using the Hydrogen basis

We use a simple system, the electron configuration in a Hydrogen-like atom, to demonstrate the importance of using a complete basis set to provide a proper quantum mechanical description. We first start with what might be considered a successful strategy --- to diagonalize a truncated Hamiltonian matrix, written in a basis consisting of Hydrogen ($Z=1$) basis states. This fails to provide the correct answer, and we then demonstrate that the continuum basis states provided the rest of the true wave function, for the bound ground states. This work then shows, in a relatively simple system, the need to utilize a complete basis set, consisting of both bound and continuum states.