Ground- and excited-state energies and wavefunctions of a hydrogen-like atom, confined at the centre of a spherical `box' with impenetrable walls, are derived using a variety of analytical and algebraic methods. In particular, asymptotic forms (which yield highly accurate energies) are obtained for the case of large box radii, and departures from the Coulomb degeneracy for a box of finite radius demonstrated. For smaller boxes, economical wavefunctions are developed on the basis of unconventional forms of Rayleigh-Schrödinger perturbation theory, and of a Lie algebraic treatment of a transformed Schrödinger equation.

We present exact analytical solutions for the much-studied problem of a hydrogen-like atom confined in a spherical box of radius R. These solutions, which are obtained for all states and all R, are expressed directly in terms of the Kummer M-functions whose analytical and numerical properties are well known, and may be calculated using standard computing packages. The solutions are illustrated by precise calculations that yield accurate energies E for any given radius R, or for R when E is known. In the special case where E ϭ 0, it is shown that the solution may be expressed in terms of Bessel functions. Finally, the physical assumptions made in applying this model to describe atomic confinement are discussed critically.

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