Quantum mechanics and the hydrogen atom in a generalized Wigner–Seitz cell

We investigate the energy spectrum of a nonrelativistic quantum particle and a hydrogen-like atom placed in a vacuum cavity with general boundary conditions ensuring confinement. When these conditions, as in the Wigner–Seitz model, admit a large amplitude of the wave function on the boundary of the cavity, a nonperturbative rearrangement of lower energy levels of the spectrum occurs, which is essentially different from the case of the confinement by a potential barrier. A nontrivial role in this spectrum rearrangement is played by the von Neumann–Wigner effect of repulsion of nearby levels. For such a confined state of a hydrogen-like atom in a spherical cavity of radius $R$ with the boundary formed by a potential layer of depth $d$, we show that the lowest energy level of the atom has a pronounced minimum at physically meaningful layer parameters and that the binding energy can be much greater than $E_{1s}$, the energy of the 1s level of a free-standing atom, and that the regime where the atom binding is much greater than $E_{1s}$ becomes possible for a cavity with $R\sim10$–$100$ nm.