Nonlinear Schrödinger–Poisson theory for quantum-dot Helium

We use a nonlinear Schrödinger–Poisson equation to describe two interacting electrons with opposite spins confined in a parabolic potential, a quantum dot. We propose an effective form of the Poisson equation taking into account the dimensional mismatch of the two-dimensional electronic system and the three-dimensional electrostatics. The results agree with earlier numerical calculations performed in a large basis of two-body states and provide a simple model for continuous quantum–classical transition with increasing nonlinearity. Specific properties due to eigenstate non-orthogonality are emphasized.