Coupled-Equations Method for the Scattering of Identical Particles

The completely antisymmetric solution ${\Psi }^A$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\Psi }^A=\Sigma {\varphi }_{\alpha }{\psi }_{\alpha }$, where $\left\{{\varphi }_{\alpha }\right\}$ is a complete set of antisymmetric states for the target and ${\psi }_{\alpha }$ are one-particle functions. Coupled equations for the ${\psi }_{\alpha }$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\Psi }^A$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\psi }_0$ is shown to obey an inhomogeneous differential equation, rather than a Schrödinger equation. The homogeneous solution is identical to the elastic wave function obtained when the projectile is distinguishable, while the inhomogeneous solution is due entirely to exchange effects. The function ${\psi }_0$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\psi }_0$ obeys a Schrödinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\psi }_0$ yields the exact elastic amplitude including direct and exchange contributions, and a phase-shift analysis of the exchange term is given. The more standard form of solution ${\Psi }^A=\Sigma \mathcal{a}\{{\varphi }_{\alpha }{f}_{\alpha }\}$, where $\mathcal{a}$ is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.