Particles that must have symmetric states are called bosons; pions, photons, and He 4+ nuclei are such particles. Particles whose states must be ant symmetric under interchange of the two particles are called fermions; electrons, protons, and neutrons are examples of fermions. Particles with integer spin are always bosons while particles with half-integer spin are always fermions. It is not at all obvious from the point of view of non-relativistic quantum mechanics why this relation, called the connection between spin and statistics, is true, and one must look within the framework of relativistic quantum field theory to understand it fully theoretically. Effects of identity can also show up in the scattering of seemingly non-identical particles. The elastic scattering of a He 4+ ion from a He 4+ atom, both are in their ground states. These are certainly distinguishable particles, since they have different charges. Particle identity also plays an important role in determining the properties of molecules made of identical atoms.
The completely antisymmetric solution ${\ensuremath{\Psi}}^{A}$ to the problem of the scattering of a fermion by a finite system of identical fermions is studied by means of the expansion ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}{\ensuremath{\phi}}_{\ensuremath{\alpha}}{\ensuremath{\psi}}_{\ensuremath{\alpha}}$, where ${{\ensuremath{\phi}}_{\ensuremath{\alpha}}}$ is a complete set of antisymmetric states for the target and ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are one-particle functions. Coupled equations for the ${\ensuremath{\psi}}_{\ensuremath{\alpha}}$ are found that obey the proper boundary conditions. This is done by means of the integral equation for ${\ensuremath{\Psi}}^{A}$ and the use of projection operators. The elastic-scattering (optical-model) wave function ${\ensuremath{\psi}}_{0}$ is shown to obey an inhomogeneous differential equation, rather than a Schr\"odinger 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 ${\ensuremath{\psi}}_{0}$ is identical to the optical-model wave function found by Bell and Squires, who showed that ${\ensuremath{\psi}}_{0}$ obeys a Schr\"odinger equation with an optical potential containing direct and exchange contributions. It is shown that ${\ensuremath{\psi}}_{0}$ 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 ${\ensuremath{\Psi}}^{A}=\ensuremath{\Sigma}\mathcal{a}{{\ensuremath{\phi}}_{\ensuremath{\alpha}}{f}_{\ensuremath{\alpha}}}$, where
a is an antisymmetrizer, is briefly discussed. The extension to the cases of inelastic scattering and deuteron elastic scattering is made.