Hydrogenic atom in semi-infinite space

A variational method is developed to determine the recycling function in a Green's-theorem calculation of the energy levels of a hydrogenic atom in semi-infinite space. It is shown that, with suitable choice of the trial wave function, the problem is reduced to an equivalent one-dimensional eigenvalue problem with eigenvalue E, the energy shift of the bulk levels. The method can be applied to any state, and the three lowest energy levels obtained this way are compared with available existing results. The change of eigenstates as the atom approaches the surface is also discussed.