Although the relative motion and that of the centre of mass cannot in general be separated for an exciton in a quantum well, there is an amusing special case where this is possible. Let Ve and Vh both be parabolic with the same frequency, so
Show that separation into RCM and R is now possible. Unfortunately one still has to solve the equation involving the relative coordinate with both the parabolic potential and the interaction. Sometimes the further approximation is made that the interaction is parabolic in R, rather than 1/R; show that this makes the problem simple.
The results are more useful when applied to an electron (or hole) gas in a parabolic potential, because they show that the motion of the centre of mass is independent of interactions between the electrons. This is known as Kohn theorem and is important in determining the infrared response of doped quantum wells, dots, or superlattices.