On the theory of elastic collisions between electrons and hydrogen atoms

A hydrogen atom in the ground state scatters an electron with kinetic energy too small for inelastic collisions to occur. The wave function $\psi$ (r$_1$, r$_2$) of the system has boundary conditions at infinity which must be chosen to allow correctly for the possibilities of both direct and exchange scattering. The expansion $\psi = \sum_\gamma \psi_\gamma (r_1)F_\gamma(r_2)$ of the total wave function in terms of a complete set of hydrogen atom wave functions $\psi_\gamma$(r$_1$) includes an integration over the continuous spectrum. It is shown that the integrand contains a singularity. The explicit form of this singularity and its connexion with the boundary conditions are examined in detail. The symmetrized functions $\Psi^\pm$ may be represented by expansions of the form $\sum_\gamma {\psi_\gamma(r_1)G^{\pm}_\gamma(r_2)\pm\psi_\gamma(r_2)G^{\pm}_\gamma (r_1)},$ where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states $\psi_\gamma$ of the hydrogen atom are included in the expansion, the equation satisfied by F$_1$, the coefficient of the ground state, contains a polarization potential which behaves like -$\alpha$/2r$^4$ for large r and is independent of the velocity of the incident electron.