  A full strength Coulomb interaction between trapped electrons can be felt
only in absence of a neutralizing background. In order to study quantum
degenerate electrons without such a background, an external trap is needed to
compensate for the strong electronic repulsion. As a basic model for such a
system, we study a trapped electron pair in a harmonic trap with an explicit
inclusion of its Coulomb interaction. We find the eigenenergy of the ground
state, confirming earlier work in the context of harmonium. We extend this to a
complete set of properly scaled energies for any value of the trapping
strength, including the excited states. The problem is solved either
numerically or by making harmonic approximations to the potential. As function
of the trapping strength a crossover can be made from the strongly to the
weakly-coupled regime, and we show that in both regimes perturbative methods
based on a pair-wise electron description would be effective for a
many-particle trapped electron system, which resembles a Wigner crystal in the
ground state of the strongly coupled limit.
