  A simple position probability density formulation is presented for the motion
of a particle in a spherically symmetric potential. The approach provides an
alternative to Newtonian methods for presentation in an elementary course, and
requires only elementary algebra and one tabulated integral. The method is
applied to compute the distributions for the Kepler-Coulomb and isotropic
harmonic oscillator potentials. Formulas are also deduced for the average
values for powers of the radial coordinate, and applied to describe
perturbations to these systems. The classical results are also compared with
quantum mechanical calculations using the Einstein-Brillouin-Keller
semiclassical quantization.
