File: Error cancellation in the semiclassical calculation of the scattering length

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M.J. Jamieson, and H. Ouerdane

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 3Σ+ 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 Χ1Σ+ and a3Σ+ molecular potentials and we use it to demonstrate that the semiclassical formula fails for cold collisions of H atoms in the Χ1Σ+g molecular state because of the long range errors rather than because of inadequacies in describing the motion over the potential well semiclassically.

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