Radial Schrödinger equation for a central potential ($-\frac{{\hslash }^2}{2m}(R^{\prime \prime }+2R^{\prime }/r)+[V(r)+{\hslash }^{2}l(l+1)/(2m{r}^2)]R=ER$)

For a spherically symmetric potential energy $V(r)$, separation of variables with $\psi =R(r)Y_{lm}(\theta ,\varphi )$ reduces the Schrödinger equation to the displayed radial equation. Writing $u=rR$ instead removes the first derivative:The angular contribution acts as a centrifugal barrier. Boundary behavior and wavefunction normalization must still be imposed; an arbitrary radial function is not an energy eigenfunction.