Applications of exact solutions for confined systems

Models for atomic hydrogen and hydrogen-like atomic systems are well known and have been utilized in many physical applications. The main interest in this proposal are similar systems where the electron is confined to a definite spherical region. (It is also proposed to examine non-spherical regions of confinement). The exact solution for spherically confined hydrogen has been discovered and rediscovered over many years and may be expressed quite simply in terms of standard mathematical functions ( the Kummer M functions). Earlier work was mainly formal due to the difficulty in evaluating these functions numerically but the present situation is very different and there have been extensive analytic and numerical studies of these functions and they may also be evaluated efficiently using modern symbolic mathematical software. I have been a co-author of recent work demonstrating this and extending the analytical solutions to deal with the case where the atom is perturbed. These time-independent perturbations may be used may be used to model effects of fields and many other physical effects and our work showed that exact solutions can be obtained for perturbed confined systems that reduce to the well known solutions in the unconfined situation.The applications of the confined systems are diverse and include models for the effect of pressure on energy levels,ion traps, the cell model of a liquid state, semiconductor quantum dots and problems in astrophysics such as the rate of escape of stars from galactic clusters and the theory of white dwarf stars. In this proposal we will seek exact or approximate analytic solution to the equations that describe these problems. It will involve extending the models so that the potential is different in different regions and treating time-dependent perturbations. In this proposal we also wish to enhance a new collaboration where the confined atomic hydrogen wavefunctions are used to estimate the exchange energy between two spatially separated hydrogen atoms. In this new theory the wavefunction on just one of the atoms can be used to estimate the two centre effect, but it is necessary that it is confined. ( Previous calculations have just ignored the effect of the wavefunction outside the significant region). This problem is of wide interest in molecular physics especially in astrophysical applications and the most useful outcome would be an approximation in terms of the distance between the two atoms. It is also planned to develop similar model potentials to treat other atoms.