7.2o9 5.635 4.586

momentum the larger the spatial extent and therefore, the larger the distance between the electrons [186]. To confirm this numerically, we list in Table 6 the roots r0 of the potential for the interacting electrons in the quantum dot, for states with different angular momenta. At particular values of the ratio ac/a0, as the azimuthal quantum number mi increases, the root r0 also increases and thus the electron-electron interaction Vee(r) = 2/r0, in the leading term of the energy series expression, decreases.

In Figure 68, we showed the dependence of the ground state energy on the magnetic field strength for confinement energies: ha0 = 6 and 12 meV. For constant values of the magnetic field, the larger the confinement energy, the greater the energy of the interacting electrons in the quantum dot. The spin effect can be included in the Hamiltonian, Eq. (316), added to the center of mass part as a space independent term, and Eq. (318) is still an analytically solvable harmonic oscillator Hamiltonian [180].

We have compared, in Table 7, the calculated results for the ground state energies |00> of the relative Hamiltonian at different confining frequencies with the results of Taut [187]. In a very recent work, Taut has reported a particular analytical solution of the Schrodinger equation for two interacting electrons in an external harmonic potential. The table shows that as 1/a increases the difference between both results noticeably decreases until it becomes ~1.4 x 10-3 at 1/a = 1419.47.

Quantum dots with more than two electrons can also be studied. The Hamiltonian for ne-interacting electrons, provided that the electron-electron interaction term depends only on the relative coordinates between electrons V(\rt — rj|) = e2/K\rj|, and parabolically confined in the quantum dot, is separable into CM and relative Hamil-tonians. The parabolic potential form V(ri) = mia2rf/2, i = 1, 2, 3,.. .,ne, is the only potential which leads to a separable Hamiltonian. The CM motion part is described by

Table 7. The ground state energies (in atomic units) of the relative Hamiltonian calculated by 1/N expansion at different frequencies, compared with the results of Taut [1871.

1/N expansion


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