Cylindrically confined $H$ atom in magnetic field: variational cut-off factor

In the present study, we consider the hydrogen atom confined within an impenetrable infinite cylindrical cavity of radius $ρ_{0}$ in the presence of a constant magnetic field ${\bf B} = B\,\hat{\bf z}$ oriented along the main cylinder's axis. In the Born-Oppenheimer approximation, anchoring the nucleus to the geometric center of the cylinder, a physically meaningful 3-parametric trial function is used to determine the ground state energy $E$ of the system. This trial function incorporates the exact symmetries and key limiting behaviors of the problem explicitly. In particular, it does not treat the Coulomb potential nor the magnetic interaction as a \textit{perturbation}. The novel inclusion of a variational cut-off factor $\big(1 - \big(\fracρ{ρ_0}\big)^ν\big)$, $ν\geq 1$, appears to represent a significant improvement compared to the non-variational cut-off factors commonly employed in the literature. The dependence of the total energy $E=E(ρ_0,\,B)$ and the binding energy $E_b=E_b(ρ_0,\,B)$ on the cavity radius $ρ_0 \in [0.8,\,5] \,$a.u. and the magnetic field strength $B\in [0.0,\,1.0]\,$a.u. is presented in detail. The expectation values $\langle ρ\rangle$ and $\langle|z| \rangle$, and the Shannon entropy in position space are computed to provide additional insights into the system's localization. A brief discussion is provided comparing the 2D and 3D cases as well.