The corresponding Schrodinger equation is still separable, but the resulting radial equation

must be solved with the following boundary condition:

This means that, in order to obtain the energy spectrum, we must find the roots of Eq. (306). The new situation must be tackled in a more complicated way since we must construct a convergent series representation of R(r) and then solve it numerically.

Alternatively, we solve the same problem approximately, with the aid of the modified variational method discussed previously. We exploit the fact that the solutions of the "free" hydrogen atom are known [see Eq. (302)] in order to choose the trial wavefunctions as

Xnm(r, 0, V) = N(r0 - r)(2ar)lF(-n + l + 1, 2l + 2, 2ar)

x expi-ar)YlmiO,p)

where N is a normalization constant that depends on r0 and a, as well as on n and l.

We can note that 1/n is replaced by a in this choice for X'. The reason is that, as a result of confinement, the number n is no longer a good quantum number to specify the state of the system (l is still a good quantum number since symmetry has not been broken). This ansatz gives more flexibility to the variational wavefunctions, allowing for the calculation not only of the ground state energy, but also of excited states, with only one variational parameter. Furthermore, the quantum virial theorem for enclosed systems [165] is satisfied by these functions.

In order to show the adequacy of the method, in the following we shall restrict ourselves to states described by nodeless wavefunctions involving different symmetries.

When we use Eq. (307) as a trial wavefunction, together with the Hamiltonian given by Eq. (303), the energy can be readily found by minimizing the functional f X/e HX dT

with respect to a, within the bounded volume O, restricted to satisfy the constraints implied by o

This procedure is straightforward and can be done through direct algebraic manipulation.

Figure 63 shows the results obtained after minimizing Eq. (308) as compared to the "exact" values obtained by numerically solving Eq. (306) [124] for the ground state and the 2p and 3d excited states. Also shown in this figure are the results due to Fernandez and Castro [166] who used a method based on both hypervirial theorems and perturbation theory [167]. A remarkable agreement can be noticed, showing that the problem can be tackled in the simpler

it 1

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