The Potential

We wish to determine the eigenfunctions $\Psi({\bf x})$ and possible energy eigenvalues $E$ for a spherical symmetric potential

$$V({\bf x})=V(r),\quad r=\vert{\bf x}\vert.$$ (280)

An example for such a potential is the potential energy of the Coulomb attraction between a fixed point charge $Ze>0$ sitting at the origin ${\bf x=0}$ and a moving particle of mass $m$ and charge $-e$ at a distance $r$ from the origin,

$$V(r)= -\frac{Ze^2}{4\pi\varepsilon_0 r}.$$ (281)