Numerical integration of certain differential equations encountered in atomic collision theory

Abstract

The problem of accurate numerical integration of the general "two-state" coupled first-order equations frequently encountered in atomic collision theory is discussed. In particular, it is shown that an asymptotic expansion of the exact solution, valid for times early in the collision where the coupling is still weak, can be obtained by a simple iterative procedure. This expansion can be used to significantly shorten the range of numerical integration, by providing reliable starting values at finite times t₀ where the coupling is not negligible though still weak. It is further shown that spurious small oscillations in the numerical solutions, which are generated when the usual boundary conditions for t → −∞ are used to provide starting values at finite times t₀, can be removed by using the expansion for starting values at t₀, with resulting improvements by several orders of magnitude in speed and accuracy of the numerical integration. This improvement is illustrated for an example where the coupling interaction has an especially long range.