Estimating the Accuracy of the Variational Energy: The Hydrogen Atom in a Magnetic Field as an Illustration

For a hydrogen atom subject to a constant magnetic field, we report a numerical realization of the two-dimensional Non-Linearization Procedure (NLP) to estimate the accuracy of the variational energy associated with a given trial function. Relevant equations of the NLP, which resemble those describing a dielectric medium with a space-dependent permittivity and charge distribution, are solved numerically with high accuracy by using an orthogonal collocation method. As an illustration, we consider three different trial functions including the one proposed in Phys. Rev. A 103, (2021) 032820. For a magnetic field strength of $\gamma =1$ a.u., we establish the accuracy of the variational energy as well as the local accuracy of the trial function. Additionally, the accuracy of the cusp parameter and the quadrupole moment, found by means of the trial function, is investigated