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r should be solved with the condition

which would force us to use a numerical treatment. In the modified variational method (as in the case of the confined hydrogen atom) the symmetry of the enclosed harmonic oscillator is not broken by the confinement and l is still a good quantum number but n ceases to be so. To construct the trial wavefunctions, we simply replace r2 by ar2 in the argument of the functions in Eq. (311) and we proceed, as in our previous example, to find the energy of the system for each box size.

In Table 5 we compare the exact results obtained from Eq. (315) [124] and those obtained by the application of the variational procedure, for the ground state and the first excited state. For completeness, we also show the results for the ground state, obtained through the hypervirial pertur-bational method using an 11-term perturbation expansion [168]. Once again, a very good agreement is observed.

An important criterion for this selection is the fulfillment of the virial theorem for these systems [165]. At this stage it is important to note that there are several papers in the literature dealing with different techniques to tackle these problems [125].

In this connection, Fernandez et al. [165, 169-171] have done a thorough study of the use of the virial theorem for quantum systems subject to Dirichilet and/or Neumann boundary conditions. As we have mentioned before, in both examples, the agreement of the results obtained by the proposed modification of the direct variational approach is remarkable, in spite of the simplicity of the method.

We note that the symmetry of the systems and their confinement were intentionally chosen to be compatible (i.e., both being of spherical symmetry).

4.4. Energy States of Two Electrons in a Parabolic Quantum Dot in Magnetic Field

The energy spectra of two interacting electrons in a quantum dot confined by a parabolic potential in an applied magnetic field of arbitrary strength are obtained in this section. The

Table 5. Energy levels of the enclosed harmonic oscillator as a function of the radius of the box.

Ground state

First excited state

Table 5. Energy levels of the enclosed harmonic oscillator as a function of the radius of the box.

Ground state

First excited state

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a

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^ exact

a

^var

^ exact

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o.131o

5.o756

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