## 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.

### 1 Introduction

We consider a hydrogen-like atom of nuclear charge

where

Chaudhuri and Coon [1] treated a more general version of this problem in which the effective mass of the electron is also different in the medium outside the cavity and provided an exact analytic solution. Von Roos [2] pointed out that the Hamiltonian with position-dependent mass is not Hermitian and hence should be abandoned. Still it has been widely and successfully used in semiconductor physics particularly in quantum well problems. Considerable interest and work on the position-dependent effective mass have continued (see ref. [3] and references therein). It is not clear how to treat the position-dependent effective mass in the oscillator representation method (ORM) and it is not so important for our purposes either. So we have not included the position-dependent effective mass in the problem treated in this article. It should also be noted that the second term in

It is clear from a simple inspection of the potential that in the limits of

In Section 2, we reproduce the solution to the Schrödinger equation (SE) with the potential in equation (1) provided in ref. [1]. In Sections 3 and 4, we outline the general principles of the ORM developed by Dineykhan and Efimov [4] and present the results of the application of the ORM to the current problem, respectively. In Section 5, we compare and discuss the results obtained by the two methods for different parameters.

### 2 Cavity model: analytic solution

The SE with the potential given in equation (1) is spherically symmetric. Consequently, the wavefunction can be separated in spherical coordinates and written as^{[1]}

where

where

The constants *A* and *B* are obtained by matching the wavefunctions at

and

Equations 3(a) and 3(b) are confluent hypergeometric equations, each of which has two independent solutions. To obtain a physical solution, the solutions of equations 3(a) and 3(b) are selected so that they are regular at
^{[2]} by matching the logarithmic derivative of the radial wavefunctions corresponding to the two selected solutions appropriate for the two regions:

The functions
_{1}F_{1} and _{2}F_{0}, respectively. While the series representation of the function

### 3 ORM

Dineykhan and Efimov [4] developed the ORM arising from ideas and methods of the quantum field theory. Using the ORM they calculated the binding energies of a number of systems with various types of potentials including the Coulomb and power-law potentials, exponentially screened Coulomb potential, logarithmic potential [4,5], and a two-electron quantum dot in a magnetic field [6]. The ORM results for the Coulomb and power law, the exponentially screened Coulomb, and logarithmic potentials agree very well with the results obtained by variational numerical methods. Amin and El-Asser have applied the ORM to calculate the energy spectrum of hydrogen-like atoms in a van der Waals potential [7].

The first key step in the ORM is a transformation of the variables in the SE such that the wavefunction takes a Gaussian asymptotic form. Schrödinger in a paper [8] on solving eigenvalue problems by factorization pointed out the existence of such a transformation in which the Kepler problem is transformed into an oscillator problem in four dimensions. The modified SE in the new expanded space having the Gaussian asymptotic solution exhibit oscillator behavior at large distances.

In the next steps, the canonical variables (coordinate and momentum) in the transformed space are represented in terms of creation and annihilation operators
^{†} and
*normal ordered products* over
^{†} and
*normal order* operation (also known as Wick’s transform) is an operation in which all the creation operators
^{†} are moved to the left and all the annihilation operators
*normal order* operation is frequently used in quantum field theory. Wick’s transform :
*n*th order Hermite polynomial in
*ω*, is extracted from the Hamiltonian written in the form

which determines

Another interesting aspect of the ORM is that the dimension of the hyperspace can be a variational parameter and as a result can be non-integer. Even though an integer dimension is used to derive the energy equations, the dimension appears in the end results just as a parameter and thus can be varied to obtain the energy minimum.

Dineykhan and Efimov [4] used the following transformations that provide the Gaussian asymptotic wavefunction behavior:^{[3]}

The radial part of the SE in the new variable

where

It should be noted here that

The energy spectrum

and it is determined by

where

If the potential

The oscillator representation is then obtained by writing the Hamiltonian

and by introducing the usual creation and annihilation operators,
*q*
_{
j
} and *p*
_{
j
} (*j* = 1, 2,..., *d*), as:

The creation and annihilation operators satisfy the standard commutation relation

After normal ordering the products over
^{†} and

where

As mentioned earlier, the symbol :*: represents the *normal ordering* of the products over
^{†} and

The energy spectrum is then obtained by calculating the contribution of the interaction part

See ref. [4] for the details of the calculation of the radial eigenstates

### 4 Cavity model: ORM solution

In this section, we will apply the ORM to the cavity model defined by the potential in equation (1). We note that the calculation of the energy spectrum using the wavefunction matching method is a natural one when the potential is discontinuous. However, to apply the ORM to the cavity model it may be useful for the calculations of various terms to write the potential in a continuous form as follows:

where

Now we make the typical approximation in which there is no repulsive character in the potential by setting

where

and

In order to obtain the energy spectrum we use the ORC in equation (8) with

With a solution for

The energy spectrum given by equation (33) is still a function of the parameter

in which

An inspection of equations (31b) and (31d) indicates that

In the limiting case of

In some situations, fixing the parameter

As mentioned earlier,

### 5 Results and discussion

First, we consider the ground state, i.e., the 1s state for which

### Figure 1

As seen from the analytic results the shallow to deep transition of the energy level occurs near

Now in order to eliminate the

### Figure 2

For all states for which

### Figure 3

The solid lines in Figure 3(a and b) represent the analytic results. The 2s radial probability function has two lobes with maxima at

The ORM results with

From this analysis, it appears that at least for the ground state in which there is no contribution from the interaction part of the Hamiltonian,

Dineykhan and Nazmitdinov [6] provided analytical results for a two-electron quantum dot in a magnetic field with a harmonic oscillator potential for the quantum dot using the ORM. The problem of a quantum dot in a magnetic field was treated using numerical method [10,11] and analytical method with some approximations [12]. A singlet to triplet state transition was predicted for the ground state in a quantum dot as a function of the applied magnetic field. These transitions were experimentally observed by Ashoori et al. [13]. The harmonic oscillator potential has been used [10,11,13] to represent the confining potential in a quantum dot. However, a quantum dot having a finite size, a finite potential should provide a better model. Ashoori et al. conjectured that the discrepancy of their experimental values of the magnetic fields at which the singlet to triplet transition occurs with the theoretical values to the strictly harmonic confining potentials used in the theoretical calculations. Recently, Chaudhuri [14] obtained analytic expressions for the energy levels of a 2-electron system in a 2-dimensional quantum dot modeled with a finite Gaussian potential and subjected to a magnetic field using an ORM approach like the one presented in this article. In the appropriate limits, the results are shown to match very well with previous analytical and numerical results. Using the expressions, the magnetic field at which the spin-singlet to spin-triplet ground state crossing occurs is calculated. The calculated value is closer to the experimental value compared with the infinite harmonic potential results in previous theoretical models.

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**Received:**2020-01-28

**Revised:**2020-10-02

**Accepted:**2020-10-22

**Published Online:**2021-03-05

© 2021 Sid Chaudhuri, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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