## Usy

r0 (Bohrs)

Figure 57. (a) Region of circular confinement. Limiting case: Q-zero-dimensional. (b) Ground state energy of the two-dimensional hydrogen atom confined within a region of the plane limited by a circle, as a function of its radius r0. 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 forbidden region increases, confining the system in the —y-axis, thus realizing the transition from a confined two-dimensional system to a confined quasi-one-dimensional system, whose energy approaches zero (see Fig. 56b).

In Figure 57b, the ground state energy of the two-dimensional hydrogen atom confined in a circular region of the plane is depicted as a function of r0 (radius of the circle in atomic units). As can be observed, for r0 ^ ro, the allowed space approaches the whole plane (i.e., the system becomes the unconfined two-dimensional hydrogen atom whose ground state energy is —2 Hartrees). Another observation is that, when r0 ^ 0, the forbidden circular region becomes a point and a transition of the two-dimensional system to a quasi-zero-dimensional system (see Fig. 57a) with energy s ^ ro occurs.

Figure 58b displays the ground state energy of the two-dimensional hydrogen atom confined in a region of the plane limited to a cone obtained when 0 = 00 and 0 < r < ro, as a function of 00 (polar angle in radians). In the limit when 00 ^ 0 the cone decreases; then the transition of the confined two-dimensional system to the quasi-one-dimensional system with energy s ^ 0 occurs, as in the previous situations. When 00 increases to 2^, the allowed region grows

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