Spherically confined hydrogen atom: variational cut-off factor

The hydrogen atom confined within an impenetrable spherical cavity of radius R under the influence of a uniform constant magnetic field B is considered. For the ground state, using the variational method, we employ a physically meaningful trial wavefunction characterized by three variational parameters, including a novel cut-off factor that acts as an additional degree of freedom in the optimization process. This approach allows us to systematically analyze the interplay between quantum confinement and the external magnetic field, providing insights into their combined effects on the energy spectrum and wavefunction behavior. Our results reveal how the ground state energy E and eigenfunction evolve as functions of the cavity radius R ∈ [1, 5] a.u. and magnetic field strength B ∈ [0, 1] a.u., offering a deeper understanding of quantum confinement in atomic systems subjected to external fields. These findings have potential implications for confined quantum systems in astrophysical and nanotechnological applications.