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 Ψ(r1; r2) 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 Ψ = Σ ψ,(r1)Fy(r2) of the total wave function in y terms of a complete set of hydrogen atom wave functions ψy(r1) includes an integration over the continuous spectrum. It is si own 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 Y* may be represented by expansions of the form Σ {ψy(r1) Gy±(r2) ±ψy(r2) y Gy±(r1)}, where the integrand in the continuous spectrum does not involve singularities. Finally, it is shown that because all the states ψy of the hydrogen atom are included in the expansion, the equation satisfied by F1, the coefficient of the ground state, contains a polarization potential which behaves like — a/2r4 for large r and is independent of the velocity of the incident electron.