On the treatment of $\ell$-changing proton-hydrogen Rydberg atom collisions

Abstract

Energy-conserving, angular momentum-changing collisions between protons and highly excited Rydberg hydrogen atoms are important for precise understanding of atomic recombination at the photon decoupling era, and the elemental abundance after primordial nucleosynthesis. Early approaches to $\ell$-changing collisions used perturbation theory for only dipole-allowed ($\Delta \ell=\pm 1$) transitions. An exact non-perturbative quantum mechanical treatment is possible, but it comes at computational cost for highly excited Rydberg states. In this note we show how to obtain a semi-classical limit that is accurate and simple, and develop further physical insights afforded by the non-perturbative quantum mechanical treatment.