A solution to the problem of a hydrogenic atom in a homogeneous dielectric medium with a concentric spherical cavity using the oscillator representation method (ORM) is presented. The results obtained by the ORM are compared with a known exact analytic solution. The energy levels of the hydrogenic atom in a spherical cavity exhibit a shallow-deep instability as a function of the cavity radius. The sharpness of the transition depends on the value of the dielectric constant of the medium. The results of the ORM agree well with the results obtained by the analytic solution when the shallow-deep transition is not too sharp (i.e., when the dielectric constant is not too large) for all values of the cavity radius. The ORM results in the zeroth order approximation diverge significantly in the region of the shallow-deep transition (i.e., for the values of the radius where the shallow-deep transition occurs) when the dielectric constant is high and as a result the transition is sharp. Even for the sharp transition, the ORM results again agree very well with the analytic results at least for the ground state when a commonly used approximation in the ORM is removed. The ORM methodology for the cavity model presented in this article can potentially be used for two-electron systems in a quantum dot.

中文翻译：

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腔中氢原子的问题：振荡器表示解与解析解

提出了使用振荡器表示法（ORM）的同心球面均质电介质中氢原子问题的解决方案。将ORM获得的结果与已知的精确分析解决方案进行比较。球形腔中氢原子的能级表现出随腔半径而变的浅深度不稳定性。过渡的清晰度取决于介质的介电常数的值。对于腔半径的所有值，当浅深过渡不太明显时（即，介电常数不太大时），ORM的结果与解析解获得的结果非常吻合。ORM导致零阶近似在浅深过渡区域显着发散（即，介电常数较高时，则发生较深的跃迁时的半径值，结果跃迁变陡。即使对于急剧的过渡，ORM结果也再次与分析结果非常吻合，至少在删除了ORM中常用的近似值时对于基态而言。本文介绍的腔模型的ORM方法可以潜在地用于量子点中的双电子系统。