Exact solutions for shell-confined hydrogen-like atoms: Polarisabilities and Shannon entropies

An idealised model to treat the effect of spherically confining the electron in a hydrogen-like atom is studied, where the potential is infinite in all space except for a spherical shell. The exact solution of the Schrdinger equation is obtained in terms of two independent solutions of the Kummer equations. It is found that, in some cases, it is necessary to use the standard Kummer M function and a non-standard second solution. In other cases we may use the Kummer U function and in a limiting case the two standard solutions of Bessel's equation. The effect of an imposed dipole field on the shell is treated using the first-order perturbation equation from which the polarisability can be calculated. In addition, the exact wavefunction is used to calculate the Shannon entropies of both position and momentum and it is shown that these measures give insight into the form of the wavefunction.