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A hydrogen-like atom confined within an impenetrable spherical box

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.