Hydrogen-like Atoms

This chapter shows how the electronic Schrodinger equation (SE) is solved for a hydrogen-like atom, i.e. an electron moving in the field of a fixed point-like nucleus with charge number Z. The hydrogen atom corresponds to Z = 1. The potential in atomic units is –Z/r, with r being the distance of the electron from the nucleus. The SE is not separable in Cartesian coordinates, but in spherical polar coordinates it separates into a radial equation and an angular momentum equation. The bound states have a total energy of –Z2/(2n2), with n = nr + ℓ being the principal quantum number (q.n.), ℓ = 0,1,2,… the angular momentum q.n., and nr = 1,2,3,… being a radial q.n. Each state for a given ℓ is 2ℓ+1-fold degenerate, with the components labelled by the projection q.n. mℓ. The wavefunctions for mℓ ≠ 0 are complex, but real linear combinations can be formed. This gives the atomic orbitals known from general and organic chemistry. Different ways of visualizing the real wavefunctions are discussed, e.g. as iso-surfaces.