Figure 56. (a) Region of parabolic confinement. Limiting case: Q-one-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a parabola, as a function of £0. 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.

branches coincide with the y-axis, so that the confinement region becomes a half plane for any value of a different from zero and infinity. When and a ^ 0, the coalescence of the two branches of the hyperbola (^0 ^ 0- and ^ 0+) leads to a quasi-two-dimensional hydrogen atom with its nucleus at the y-axis; the ground state energy approaches —2/9 Hartree. When a ^ ro, the whole plane becomes available for the atom, thus leading to the unconfined two-dimensional hydrogen atom (see Fig. 55a). Another observation is that for 1 the whole plane is forbidden (x-axis is excluded), thus resulting in the one-dimensional hydrogen atom with zero energy (see Fig. 55b), which physically means that all excited states belong to the continuous spectrum or to the ionized atom. A similar behavior was found in [142, 143] for a three-dimensional hydrogen atom limited by paraboloidal or hyperboloidal surfaces.

Figure 56b shows the energy for the ground state of the two-dimensional hydrogen atom confined in a region of the plane limited by a parabola (see Fig. 56a) as a function of (which defines the degree of confining). Notice that as ^ ro the parabola opens and the confinement region decreases until the whole plane is available; thus a transition occurs from a confined two-dimensional system to an uncon-fined two-dimensional system (see Fig. 56a), whose energy is 2 Hartrees. In the same way, if ^ 0 the parabola closes

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