Two interacting electrons in a magnetic field: comparison of semiclassical, quantum, and variational solutions

The quantum mechanical many-body problem is rarely analytically solvable. One notable exception is the case of two electrons interacting via a Coulomb potential in a uniform magnetic field. The motion is confined to a two-dimensional plane, which is commonly the case in nanodevices. We compare the exact solution with the semiclassical energy spectrum and study the time-dependent dynamics of the system using the time-dependent variational principle.