Correlation energies for two interacting electrons in a harmonic quantum dot

The problem of correlated electrons in a quantum dot is interesting and exciting. Their confinement within a harmonic potential provides not only a needed environment but also a unique signature to the physics of the dynamic correlation. The energy level structure of the relative motion of two electrons in such a quantum dot is studied in detail. Simple but accurate analytic expressions for the correlation energy are found by a double-parabola approximation in a WKB treatment, whose results are shown to be in excellent agreement with those of exact numerical solutions. Based on the analytical study, however, a clear physical picture emerges. It is found that the internal energy levels of a given angular momentum $m$ can be likened to a ladder of nearly equally spaced energy steps placed on a pedestal. The height of the pedestal is given by $V_{\min }(m,\lambda ),$ the minimum of the effective interaction potential associated with a given $m$ and the Coulomb coupling strength λ relative to the confinement. Superimposing ladders of all possible values of $m$ then yields the entire level structure for the correlated relative motion. Owing to the unique nature of harmonic confinement, correlation thus manifests itself mostly through the λ dependence of the classical-like $V_{\min }(m,\lambda ),$ in which the wave nature of the electrons plays no role except for the discrete values of $m.\mathrm{}$