Application of geometric probability techniques to the evaluation of interaction energies arising from a general radial potential

A formalism is developed for using geometric probability techniques to evaluate interaction energies arising from a general radial potential V(r₁₂), where r₁₂ =|r₂-r₁|. The integrals that arise in calculating these energies can be separated into a radial piece that depends on r12 and a nonradial piece that describes the geometry of the system, including the density distribution. We show that all geometric information can be encoded into a "radial density function" G(r₁₂;p₁,P₂), which depends on r₁₂ and the densities p₁ and p₂ of two interacting regions. G(r₁₂;p₁,p₂) is calculated explicitly for several geometries and is then used to evaluate interaction energies for several cases of interest. Our results find application in elementary particle, nuclear, and atomic physics.