Nonlinear quantization of a degenerate charged Bose gas in an external Coulomb trap

Abstract

We consider a degenerate charged Bose-Coulomb gas populating several discrete stationary boson bound states that are located in a spherical-symmetrical central Coulombian potential. Each such state is defined, through appropriate boundary conditions and normalization, by a so-called ``nonlinear eigenstate'' that is actually a solution of the coupled (linear) stationary Schr\"odinger-like Gross-Pitaevskii differential equation and the (nonlinear) Poisson equation. The corresponding eigenvalues allow us to define the energies of these degenerate boson states, much like the Koopmans orbital energy in atomic physics. This theory applies surprisingly well (compared with the corresponding Hartree-Fock results) to spherical-symmetrical $s$ orbital states in atomic physics (i.e., bosonlike restricted orbital states where the additional spin degree of freedom is already integrated out). Finally the superposition of two such stationary nonlinear eigenstates is investigated and given a semiclassical physical significance similar to a Thomas-Fermi approach. The resulting concepts apply particularly well (namely within an average 1% error bar with respect to spectroscopic data) to the $1{s}^{2}\text{\penalty1000-\hskip0pt}2{s}^{2}$ orbital states of the $3\ensuremath{\leqslant}Z\ensuremath{\leqslant}9$ atomic subsystems.