Shp

0O (Radians)

Figure 58. (a) Region of conical confinement. Limiting case: Q-two-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a cone, as a function of the opening angle 60. 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.

0O (Radians)

Figure 58. (a) Region of conical confinement. Limiting case: Q-two-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a cone, as a function of the opening angle 60. 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.

and the system behaves as a quasi-two-dimensional system; the edges of the cone are still a forbidden region (see Fig. 58a). The latter case corresponds to the minimum of the curve at 6.2856 radians (2n) where the energy is ^—2/9 Hartree and coincides with the case of hyperbolic confinement in the limit y0 ^ 1, a ^ 0 (see Figs. 55a and 58a), as expected.

Figure 59b shows the ground state energy of the two-dimensional hydrogen atom confined in a region of the plane given by the intersection of two symmetrical confocal parabolas (see Fig. 59a), as a function of (root of the distance to the foci, in atomic units). As can be observed, when ^ 0 the two parabolas (and their intersections) are closer to the origin. The allowed region is almost a point, the transition of a confined two-dimensional system to a quasi-zero-dimensional one occurs, and accordingly, the energy becomes infinite, as expected. Moreover, in the case when ^ m, the available space enlarges until the uncon-fined two-dimensional atom is recovered, that is, the energy approaches —2 Hartrees (see Fig. 59b).

The overall results show that the ground state energy of the two-dimensional hydrogen atom confined by an impenetrable potential has two extreme limiting cases, namely: (a) In the case of closed conical curves, a transition to a quasi-zero-dimensional system occurs where the ground state energy becomes infinite, that is, the confinement region approaches a point; (b) in the case of open conical curves,

0 0

Post a comment