Energy Spectra of tightly Confined Systems

Abstract

An approximate recipe to the energy eigenvalues for the problem of non-relativistic hydrogen-like atom confined in a spherical cavity of impenetrable wall is presented. The method is based on proposing an ansatz solution with a cut-off function. The asymptotic behavior of the reduced Schrödinger equation is then considered for large and small . Some approximations are applied to deform some terms of the equation, which then allowed us to apply the Nikiforov-Uvarov method in both regions. A potential parameter is introduced that plays a crucial role in the calculations. In both cases, we obtained an algebraic equation for the energy, which then can be solved for the specific potential parameters.

Fine structure in the hydrogen atom boxed in a spherical impenetrable cavity

“…Because of impenetrability a Dirichlet boundary condition is imposed on the wave function at the sphere surface. Since 1937, such a confinement model has often been utilized to simulate spatial compression on various quantum systems …”