The problem of a hydrogen atom in a cavity: Oscillator representation solution versus analytic solution

Sid Chaudhuri

doi:10.1515/phys-2021-0201

Abstract

Abstract 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.

Highlights

We consider a hydrogen-like atom of nuclear charge e in a medium of dielectric constant ε with a concentric V(r) = − e2 ε0 r + ε0 ε e2 ε0 r0 θ(r0 r) (1)e2 εr θ(r r0), where θ(x) is the Heaviside function which is 0 for x < 0 and 1 for x ≥ 0.Chaudhuri and Coon [1] treated a more general version of this problem in which the effective mass of the electron is different in the medium outside the cavity and provided an exact analytic solution
For the oscillator representation method (ORM), we have computed the values of E1s for the ρ = 1 case by using equations (35–37)
We have computed the values of ν for the analytic solution directly by solving equation (7) for ν

Summary

Introduction

E2 εr θ(r r0), where θ(x) is the Heaviside function which is 0 for x < 0 and 1 for x ≥ 0. Chaudhuri and Coon [1] treated a more general version of this problem in which the effective mass of the electron is different in the medium outside the cavity and provided an exact analytic solution. While this term makes the potential continuous, the electric field is still discontinuous It is clear from a simple inspection of the potential that in the limits of r0 → ∞ and r0 → 0, the energy levels are identical to those of a hydrogen atom in free space and in the medium of dielectric constant ε, respectively. If the free space wavefunction, or more accurately the probability function, fits almost entirely inside the cavity, the effect of the medium outside the cavity is small.

Cavity model: analytic solution

Cavity model

Results and discussion