Quantum-mechanical and semiclassical study of the collinear three-body Coulomb problem: Inelastic collisions below the three-body disintegration threshold

Abstract

A quantum-mechanical (QM) and semiclassical (SC) study of inelastic collisions in collinear three-body Coulomb systems below the three-body disintegration threshold is presented. The QM results are obtained by solving the stationary Schr\"odinger equation in hyperspherical coordinates using the slow/smooth variable discretization method. After appropriate rescaling of the hyperspherical coordinates, an asymptotic parameter $0\ensuremath{\leqslant}h\ensuremath{\leqslant}1$ that depends only on the masses of particles and has the meaning of an effective Planck's constant for the motion in hyperradius emerges. The SC results are obtained in the leading order approximation of the asymptotic expansion in $h$. The main attention is paid to investigating how the SC and QM results converge as $h\ensuremath{\rightarrow}0$. It is shown that the overall agreement for a wide spectrum of systems and processes is surprisingly good even for $h\ensuremath{\sim}1$. However, because of interference effects the convergence is not monotonic, and the SC results may be grossly in error in the situations where a destructive interference occurs. The analysis of hidden crossings clarifies mechanisms of the nonadiabatic transitions. It is shown that if the oppositely charged particle is located between the two others, the nonadiabatic transitions occur near the top of the potential barrier via the well-known $T$ series of hidden crossings. If it is located on one end of the system, then there is no potential barrier for real values of the angular variable, but there still exists an extremum in the complex plane; the mechanism of nonadiabatic transitions in this case is called the complex $T$ series of hidden crossings.