Note that we deal with a two-electron wave function \( u(r_1,r_2) \) and two-electron energy \( E^{(2)} \).

With no interaction this can be written out as the product of two single-electron wave functions, that is we have a solution on closed form.

We introduce the relative coordinate \( \mathbf{r} = \mathbf{r}_1-\mathbf{r}_2 \) and the center-of-mass coordinate \( \mathbf{R} = 1/2(\mathbf{r}_1+\mathbf{r}_2) \). With these new coordinates, the radial Schroedinger equation reads $$ \left( -\frac{\hbar^2}{m} \frac{d^2}{dr^2} -\frac{\hbar^2}{4 m} \frac{d^2}{dR^2}+ \frac{1}{4} k r^2+ kR^2\right)u(r,R) = E^{(2)} u(r,R). $$