Eigenstate expansions in a soluble model of identical particle scattering

Solutions to the Schrödinger equation and the inhomogeneous equation for the case of two identical particles interacting with a center of force are studied. Eigenstate expansions for solving each equation are explicitly introduced and their properties discussed. The case when the interparticle interaction v₁₂ is zero is then examined; this is a completely soluble problem. The eigenstate expansion solutions for the Schrödinger and inhomogeneous equations are used to explore the means by which the correct solution is obtained. Finally, approximate solutions, obtained by truncating the eigenfunction expansions, are introduced. It is seen that both methods lead to the correct amplitude when τ₁₂ = 0, even though the approximate solution to the inhomogeneous equation does not lead, in the end, to an antisymmetric solution.