r0 (Bohrs)

Figure 63. Energy levels of the enclosed hydrogen atom as a function of the radius of the box r0. Exact and variational calculations are compared (see text). Reprinted with permission from [205], J. L. Marín et al., in "Handbook of Advanced Electronic and Photonic Materials and Devices" (H. S. Nalwa, Ed.). Academic Press, San Diego, 2001. © 2001, Academic Press.

fashion proposed here. Note that, in contrast to the hypervirial treatment, the energy values obtained by the variational method always lie as an upper bound to the exact results for all box sizes, as expected.

Incidentally, Fernandez and Castro [165] were the first to obtain the ground state energies shown in Figure 63 vari-ationally. These authors used a trial wavefunction identical to Eq. (307) while analyzing the fulfillment of the quantum virial theorem for enclosed systems by approximate wavefunctions.

We now turn our attention to the case of the enclosed harmonic oscillator following a procedure similar to the case of the enclosed hydrogen atom.

Once again, the problem of the "free" oscillator is exactly solved and the energy and wavefunctions are given by [94]

where we have set h = m = ( = 1, Y m(B,p) are the spherical harmonics, F(a,b,z) is the hypergeometric function, and N is a normalization constant.

As in the case of the hydrogen atom, when we impose the confinement, the Hamiltonian is modified to give


r0 being the radius of the confining spherical box in units of (h/mo)x/2.

Once again, if we were to solve the problem exactly, the Schrodinger equation is separable and the resulting radial equation d2 2 d l(l + 1)

0 0

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