Exact Solutions for Confined Model Systems Using Kummer Functions

We treat model systems where an electron is confined in a region of space. The particular models considered have solutions which may be expressed in terms of the Kummer functions. Both standard and non-standard Kummer functions are used in these models and a comprehensive summary of the usual and exceptional Kummer functions is given. The definition of confinement is widened to treat radial confinement in any spherical shell, including the asymptotic region and cases where the electron is confined to a lower dimension. Initially we consider the theory in K dimensional space and then give particular examples in 1, 2, and 3 dimensions. A commonly treated model is the radially confined hydrogen atom in 3 dimensions with an infinite barrier on a confining sphere so that the wavefunction is identically zero on this sphere. We have extended this model to treat a more general model of spherical confinement where the derivative of the charge density is zero on the confining sphere. It is shown that the analogous models for the radial harmonic oscillator and radial constant potentials may be treated using a generic technique.