  We present a thorough analysis of the electron density distribution (shape)
of two electrons, confined in the three-dimensional harmonic oscillator
potential, as a function of the perpendicular magnetic field.Explicit algebraic
expressions are derived in terms of the system's parameters and the magnetic
field strength to trace the shape transformations in the ground and low-lying
excited states. We found that the interplay of the classical and quantum
properties lead to a quantum shape transition from a lateral to a vertical
localization of electrons in low-lying excited states at relatively strong
Coulomb interaction with alteration of the magnetic field. In contrast, in that
regime in the ground states the electrons form always a ring type distribution
in the lateral plane. The analytical results demonstrate a good agreement with
quantum numerical results near the transition point and at high magnetic field.
