Perturbation theory for the hydrogen atom in a spherical cavity with off-centre nucleus

Abstract

The problem of a hydrogen atom in an impenetrable spherical cavity of small radius R is considered on the basis of the first-order perturbational treatment of the electron-nucleus interaction taking explicit account of the boundary condition. An approximate analytic expression is derived for the electronic energy as a function of the shift of the nucleus off the centre. The estimations obtained are in good qualitative and semiquantitative agreement with the results of the finite-difference calculations. Some general features of the energy dependences are discussed qualitatively.

Band structure of a harmonically confined electron with an impenetrable boundary

“…Hence, when an electron is confined in a harmonic trap with a sufficiently large size, it can be approximately treated as a free harmonic oscillator. It is easy and necessary to extrapolate the results to other confined systems with impenetrable boundaries and to the interelectronic motion of the trapped atoms [29]. For example, the trapped 9 Be þ ions in King's experiment [14], the trap size is x 0 0 % 2:5 Â 10 À4 m % 100=a, namely x 0 % 100, from Eqs.…”

“…Particularly, the confining sizes were taken only several times of the effective Bohr radius in Refs. [27][28][29]. In these cases, if the finite boundaries are impenetrable, Eq.…”

“…In addition, the wave functions of the confined system differ from that of the free oscillator for smaller confining sizes and approximately agree with the latter for larger confining sizes. The results can be easily extended to interelectronic motion of a trapped atom [27][28][29] and to various spatially finite systems.…”

Finite-difference calculation of donor energy levels in a spherical quantum dot subject to a magnetic field

“…A similar approach was used by Corella-Madueño et al [12] and Seddik et al [13], while Guimarães and Prudente [14] used a different variational procedure which is based on the finiteelement method. Furthermore, Changa et al [15] have dealt with the case of an off-center donor in a spherical quantum dot without the magnetic field. They found good agreement between perturbation-theory and finite-difference approaches.…”

Electronic states of hydrogen atom in tetrahedral and similar polyhedral cavities

“…For many-electron systems, one has to use the numerical procedures (see, e.g., [3][4][5][6]), but in general, the same is true for one-electron systems too, although in some special situations it is possible to give a qualitative description of the energy changes under the nucleus displacements [9,[12][13][14].…”