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.
We consider a hydrogen-like atom of nuclear charge in a medium of dielectric constant with a concentric spherical cavity of radius around the nucleus and free space dielectric constant . The following potential represents such a system:
where is the Heaviside function which is 0 for < 0 and 1 for ≥ 0.
Chaudhuri and Coon  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  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.  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 was added  to ensure the continuity of the potential at although it is not an absolute requirement that the potential must be continuous. 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 and the energy levels are identical to those of a hydrogen atom in free space and in the medium of dielectric constant , respectively. Thus, the energy undergoes a transition from a shallow to a deep level in some range of as it increases from 0 to . The magnitude of the transition is proportional to . As discussed in ref. , the reason for this transition is as follows. If the free space wavefunction, or more accurately the probability function, fits almost entirely inside the cavity, then the effect of the medium outside the cavity is small. If, on the other hand, the cavity size is such that significant portions of the free space wavefunction fall both inside and outside the cavity, then there is a competition to squeeze in and to stretch out the wavefunction by the two regions resulting in an intermediate energy value. The sharpness of the transition increases as the value of increases. This transition is akin to a second-order phase transition.
In Section 2, we reproduce the solution to the Schrödinger equation (SE) with the potential in equation (1) provided in ref. . In Sections 3 and 4, we outline the general principles of the ORM developed by Dineykhan and Efimov  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
where is the normalization constant, is the radial part of the wavefunction, and s are the spherical harmonics. The radial part of the SE in dimensionless form is written as:
where , , and is the Bohr radius. The radial wavefunction has been written in terms of and in the two regions
The constants A and B are obtained by matching the wavefunctions at and from the normalization condition. The energy eigenvalue is related to the effective principal quantum number by
and is related to by
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 and respectively. Chaudhuri et al.  provided the following energy eigenvalue condition by matching the logarithmic derivative of the radial wavefunctions corresponding to the two selected solutions appropriate for the two regions:
The functions and are confluent hypergeometric functions related to the confluent hypergeometric series 1F1 and 2F0, respectively. While the series representation of the function (see equation (15) in ref. ) is convergent for all finite values of , the series representation of has zero radius of convergence. However, an integral representation of (see equation (16) in ref. ) that satisfies the convergence requirements is used to calculate the function. The eigenvalues for a given value of the angular momentum quantum number can be calculated by locating the zeroes of the expression in equation (7).
Dineykhan and Efimov  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 . 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 .
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  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 , and the Hamiltonian is written in terms of normal ordered products over † and . The 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 are moved to the right. The normal order operation is frequently used in quantum field theory. Wick’s transform : : yields the nth order Hermite polynomial in which, of course, is the harmonic oscillator wavefunction apart from the exponential term. Wick’s transform is in fact used in quantum field theory to eliminate infinity arising from the zero-point energy. See ref.  for an exposition on Wick’s calculus. The pure oscillator part with some yet unknown frequency, ω, is extracted from the Hamiltonian written in the form , where = . In addition, a requirement is imposed that the interaction part, does not contain terms quadratic in the canonical variables so that they are completely absorbed in the oscillator part. This requirement leads to the following condition:
which determines , the oscillator frequency. This condition is known as the oscillator requirement condition (ORC).
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.
The radial part of the SE in the new variable in the expanded space is then obtained as follows:
It should be noted here that is the dimension of the hyperspace, the energy is now incorporated in the new potential and summed over the repeated index from 1 to the dimension . As mentioned earlier, is assumed to be an integer for the calculations of , , until at the end when and can be treated as non-integer variational parameters. We also note that the SE is written in dimensionless form by using the length unit as and the energy unit as
The energy spectrum of the original system is obtained from the radial excitation spectrum of the Hamiltonian of equation (11)
and it is determined by
where is the radial quantum number.
If the potential does not have a repulsive character as , is typically chosen to be zero and that leads to the equation for the dimension,
The oscillator representation is then obtained by writing the Hamiltonian in the form
and by introducing the usual creation and annihilation operators, and , respectively, in terms of the canonical variables 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 , in which all the creation operators are moved to the left and the annihilation operators are moved to the right as if they commute and some manipulation the ORM Hamiltonian is obtained as
As mentioned earlier, the symbol :*: represents the normal ordering of the products over † and . In as well as in summation over repeated indices are assumed and the notation is introduced. The function is the Fourier transform of in -dimension and given as
The energy spectrum is then obtained by calculating the contribution of the interaction part of the Hamiltonian in the perturbation approach. In the zeroth order approximation, the energy spectrum is determined by
See ref.  for the details of the calculation of the radial eigenstates and the matrix elements . In this article, we will consider only the zeroth order approximation in
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 . The second and the third terms can be treated as exponential and screened Coulomb potentials, respectively, with complex exponentials by using equation (28) for the Heaviside function. The final results are obtained by applying the operation over the functions of .
Now we make the typical approximation in which there is no repulsive character in the potential by setting and then is given by equation (17). With this approximation, we obtain after some manipulations the pertinent ORM equations for the cavity model potential given in equation (27):
and and are incomplete gamma functions defined as
In order to obtain the energy spectrum we use the ORC in equation (8) with given in equations (29) and (30). We solve the ORC equation for and plug in the expression for in equations (26) and (30) to obtain the equation for as follows:
The energy spectrum given by equation (33) is still a function of the parameter . The energy spectrum is obtained by minimizing with respect to as
in which for a given value of is determined by finding the solution to equation (32).
In the limiting case of , the solution of equation (32) yields , where is the principal quantum number, and the energy spectrum appropriately reduces to that of a free space hydrogen atom, . In the limiting case of , and the energy spectrum correctly reduces to that of a hydrogen atom in a medium of dielectric constant , i.e., . In the limiting case of , it can be shown that as the parameter tends to 1 and the energy spectrum reduces to that of the free space hydrogen atom.
As mentioned earlier, for states (i.e., 1s, 2p, 3d…).
5 Results and discussion
First, we consider the ground state, i.e., the 1s state for which In Figure 1(a–d), we have plotted as a function of for a set of parameters (0.088, 0.25, 0.5, and 0.75) obtained by the two methods presented earlier for the 1s state. For the ORM, we have computed the values of for the case by using equations (35–37). As noted earlier, even when for the state. The values of are then obtained by using equation (5). We have computed the values of for the analytic solution directly by solving equation (7) for . The value of corresponds to silicon for which the dielectric constant
As seen from the analytic results the shallow to deep transition of the energy level occurs near , where the wavefunction is maximum. Evidently, the ORM results agree very well with the analytic results for all values of when the dielectric constant of the medium outside the cavity does not differ significantly from that of the free space cavity. However, the ORM results diverge significantly in the neighborhood of where the energy level undergoes shallow to deep transition when the dielectric constant of the medium outside the cavity differs significantly from that of the free space cavity. For the realistic case of silicon ( the disagreement is stark.
Now in order to eliminate the restriction we computed the energy level by finding the minima of the energy with respect to following equations (32), (33), and (34). The dashed line in Figure 2 represents as a function of . The value of was found to vary between 1 and 2.14. As expected, the values of differ from 1 only in the transition region. With the minimization of the energy with respect to the parameter , or equivalently, the dimension , the ORM results agree very well with the exact analytic results including in the transition region even for the case of sharp transition ( ).
For all states for which that includes the ground state (1s), the contribution from the interaction part is zero since . The first state for which the contribution from the interaction part is non-zero is the 2s state. We present the results in Figure 3(a and b) for the values of and 0.5.
The solid lines in Figure 3(a and b) represent the analytic results. The 2s radial probability function has two lobes with maxima at and at Consequently, there are two sharp transitions close to these two values of with a plateau in between.
The ORM results with diverge widely in the transition region for the 2s state. So we have shown the ORM results only with that minimizes the energy. The value of was found to vary between 1 and 2.45 (i.e., for and from 1 to 1.28 (i.e., for Even with the minimization with respect to the ORM result for the sharp transition case ( ) differs significantly from the analytic result in the transition region. The origin of the discrepancy for this case can be attributed to the extremely shallow zero-crossing of the function given in equation (32), where the value of the function remains very close to zero for a wide range of values. A small change in the function would move the zero-crossing from the left to the right of the wide shallow region of . A small error in the function thus would change the calculated energy significantly. This indicates that higher order perturbation terms in the interaction part may be necessary to obtain better ORM results for sharp transition cases for states 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, , the ORM provides excellent results when the energy is minimized with respect to the parameter, . On the other hand, for states where the interaction part is non-zero, such as the 2s state, the ORM results are not so accurate in the transition region when the transition is sharp. It appears that for these cases higher order perturbation terms in the interaction contribution are necessary for more accurate results.
Dineykhan and Nazmitdinov  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 . 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. . 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  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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© 2021 Sid Chaudhuri, published by De Gruyter
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