Abstract
Exact integral expressions and simple analytical estimates for the bound-bound, bound-continuum, and continuum-continuum dipole matrix elements are derived by use of the momentum-space eigenfunctions for a one-dimensional model of a hydrogen atom. These results provide the essential ingredients for the numerical study of the quantum mechanisms responsible for the chaotic ionization of highly excited hydrogen atoms in intense microwave fields. A specific numerical algorithm for solving the Schr\odinger equation for a one-dimensional hydrogen atom in an oscillating electric field is described which uses these results for the dipole matrix elements along with a discrete representation of the continuum. In addition, the momentum-space representation of the Sturmian basis functions has been used to derive exact integral expressions and convenient analytical estimates for the projections of the Sturmian basis functions onto the hydrogenic bound and continuum states. These results are used to provide a direct comparison of numerical calculations for the ionization of one-dimensional hydrogen atoms using the hydrogenic and Sturmian bases.
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