Figure 59. (a) Region of confinement given by the intersection of two symmetrical parabolas. Limiting case: Q-zero-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by the intersection of two symmetrical con-focal parabolas, as a function of 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.

a transition to a quasi-one-dimensional system occurs in which the atom is confined to a straight line with origin at the nucleus and whose ground state energy becomes zero. Case (b) is closely related to the one-dimensional hydrogen atom discussed in [144-149], except that the binding energy is found to be infinite in [148]. The main difference between the limiting case (b) and the so-called Loudon one-dimensional atom [148] rests in the fact that Loudon did not take into account the energy due to the confinement because his model is one-dimensional and our limit case is quasi-one-dimensional. A discussion of this matter is made in next section. Moreover, since the binding energy of Loudon's one-dimensional atoms is greater than the rest mass energy of the electron, the treatment of the atom should be done in a relativistic scheme, which renders a large, but finite, binding energy [150].

In summary, the confining geometries studied in this section lead us to conclude following extreme behavior:

Circular: r

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