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Figure 61. Ground state energy for a hydrogenic impurity within quantum dots of: (a) spherical symmetry as a function of the dot radius and barrier height, (b) prolate spheroidal symmetry as a function of the eccentricity and barrier height. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001. Academic Press.

Figure 61. Ground state energy for a hydrogenic impurity within quantum dots of: (a) spherical symmetry as a function of the dot radius and barrier height, (b) prolate spheroidal symmetry as a function of the eccentricity and barrier height. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001. Academic Press.

different interfocal distances (2R). The solid curves correspond to R = 0.5 and R = 1.0 effective Rydbergs without the Coulomb term, while dashed ones are for the same values of R but the Coulomb term in the exterior of the quantum dot is included. The latter figure confirms that there is a significant variation of the energy for different interfocal distances of the spheroidal quantum dot and that the difference is greater for the lower value of R and even more significant when the Coulomb term in the exterior of the quantum dot is not included.

Thus, the previous results show that a quantitative fitting of the energy-size curves for the impurity (or, qualitatively, for the exciton) to experimental results (for a given value of V0) must be taken with caution, independently of the model used to make comparison. This is the case in most of the theoretical approaches dealing with semiconductor crystallites, since a spherical shape is assumed to fit the experimental results.

Figure 62. Ground state energy for a hydrogenic impurity as a function of the interfocal distance and the eccentricity for a fixed value of the barrier heigth. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

Figure 62. Ground state energy for a hydrogenic impurity as a function of the interfocal distance and the eccentricity for a fixed value of the barrier heigth. Reprinted with permission from [205], J. L. Marin et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

4.3. Hydrogen Atom and Harmonic

Oscillator Confined by Impenetrable Spherical Boxes

The direct variational method is used to study two simple confined systems, namely, the hydrogen atom and the harmonic oscillator within impenetrable spherical boxes. The trial wavefunctions have been assumed as the product of the "free" solutions of the corresponding Schrodinger equation and a simple function that satisfies the respective boundary conditions. The energy levels obtained in this way are extremely close to the exact ones, thus proving the utility of the proposed method.

4.3.1. Direct Variational Approach

The exact solution for the free system can be found in any text of quantum mechanics [94]. Indeed, the corresponding energy and wavefunctions are given as

$nim(r, 0, p) = Nnl(2r/n)lF(—n + l + 1, 2l + 2, 2r/n)

where h = m = 1, Yp(0, p) are the spherical harmonics, F(a,b,z) is the confluent hypergeometric function [164], and Nn is a normalization constant.

When we impose confinement on this system, the Hamil-tonian is slightly modified and can be written as

where

0 0

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