## Fiqi qi 0 q e dD

where 3D is the boundary of D.

The model Hamiltonian for the confined quantum system under study can be written, in atomic units, as

where Ven(q) and Vb(q) are given by Eqs. (265) and (266), and V2 is derived from Eq. (263).

At this point, the variational method can be implemented to calculate the ground state energy of the system by constructing the functional s(a) = f

- 2 Vq2 + Ven(q)\Xid2q and minimizing it with respect to a, restricted to f XÎXi d2q = i Jd

as usual.

The area element is given as d2q = h1h2 dq1 dq2.

To this extent, the method described previously can be used to study a confined system for which the interactions with the surrounding medium are not considered (i.e., enclosed within a potential barrier of infinite depth). The next subsection is restricted to deal with the case of a typical one-electron confined system in two dimensions, namely, the case of hydrogen atoms confined within a given region of the plane with different geometries.

4.1.2. Variational Wavefunctions for Different Geometries

As is well known, the Schrodinger equation for the uncon-fined two-dimensional hydrogen atom is separable in polar coordinates (r, 6); its ground state wavefunction can be written as

where C0 is a normalization constant, r is the electron-nucleus distance, k0 = ^J—2s0, s0 = —2 Hartrees is its ground state energy, and the nucleus was assumed to be located at the origin.

When the atom in the plane is restricted to a given open or closed domain D, $ and s0 must change to fit the new conditions accordingly. If the boundary, 3D of D, is impenetrable for the electron, then $ = 0 at 3D. The latter means that due to confinement, new quantization rules must be found; that is, the old "good" quantum numbers used to define the energy of the unconfined system are no longer useful to characterize the energy of the now confined system. Moreover, as the region of confinement was assumed arbitrary (in shape and size), the energy of a given state would depend on a continuous parameter associated with the size (or shape) of the domain D (or its boundary 3D).

In the case of the confined two-dimensional hydrogen atom, the approximate (variational) ground state energy would involve the use of a trial wavefunction given by xq qi) = Af(qh qo )exp[-ar(ql, qi)}

where (q1, q2) is the system of coordinates compatible with the symmetry of the confining boundary, r(q1, q2) is the electron-nucleus distance in these coordinates, f(q, q0) is a geometry-adapted auxiliary function such that f = 0 at qi = q0, qi is a coordinate associated with the geometry of the boundary, and a is a variational parameter.

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