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

The spectrum of the hydrogen atom confined in a spherical impenetrable box of radius R_c has been investigated by many authors up to date, but not at the level of relativistic corrections. It is well known that, as R_c diminishes, all energy levels and the pressure increase very rapidly, whereas the polarizability goes to zero. In this report, we have computed the relativistic corrections that underlie the fine structure of the confined hydrogen atom, as a function of R_c. Such corrections correspond to relativistic kinetic energy, spin‐orbit coupling and the Darwin term, which are calculated in the frame of time‐independent perturbation theory, for which, use was made of the exact confined hydrogen atom wave functions. We show that for a confinement radius of 0.5 au the relativistic corrections increase up to three orders of magnitude with respect to those corresponding to the free atom. As R_c decreases, the kinetic energy correction and the spin‐orbit coupling for become negative whereas their absolute value and the Darwin term, which is positive, increase very rapidly.