Error cancellation in the semiclassical calculation of the scattering length

Abstract

We investigate the effects of two approximations concerning long range dispersion forces that are made in the derivation of the semiclassical formula for the scattering length of a pair of neutral atoms. We demonstrate numerically, using a published model interaction potential for a pair of Cs atoms in the \(^3{\rm\Sigma}_{\rm u}^+\) molecular state, that the subsequent long range errors tend to cancel and we show, from an approximate analytical relationship, that the first order errors do indeed largely cancel. We suggest a hybrid method that combines quantum mechanical and semiclassical calculations. We explore its use in finding the scattering lengths of 7Li atoms and 133Cs atoms interacting via the X\(^1{\rm\Sigma}^+\) and a\(^3{\rm\Sigma}^+\) molecular potentials and we use it to demonstrate that the semiclassical formula fails for cold collisions of H atoms in the X\(^1{\rm\Sigma}_{\rm g}^+\) molecular state because of the long range errors rather than because of inadequacies in describing the motion over the potential well semiclassically.