Two interacting electrons in a three-dimensional parabolic quantum dot: a simple solution

We present a simple solution to the problem of two interacting electrons confined by a three-dimensional parabolic potential. The method relies on the diagonalization of the Hamiltonian in reduced Hilbert space. The basis functions are the solutions of the centre-of-mass motion and the relative motion of the two particles without Coulomb interaction. Since the Coulomb interaction only affects the relative motion, the matrix elements of the Hamiltonian are easily evaluated analytically. The numerical diagonalization is readily performed as only a few basis functions are needed to obtain a good precision on the energy levels: six basis functions ensure a precision better than 0.1% on the ground-state energy, while three basis functions are enough to obtain a precision better than 1%. The results are analysed and compared to previously published results. They are also used to evaluate the precision of a first-order perturbation calculation for the Coulomb interaction and an approach based on a 1/r(2) approximate interaction potential for which there exists an analytical solution.