Overcoming roundoff errors in the first-order amplitude for general state-to-state transitions in hydrogen by projectile impact

Abstract

A computer program to calculate the amplitude for general state-to-state transitions in hydrogen from the analytical result has the potential for significant numerical round-off errors whenever the sum of the angular momenta of the two states is greater than four and the projectile’s impact parameter is less than the ratio of its energy to velocity. This arises from high-order cancellation of terms in the MacDonald functions in that analytical result, whose polynomial portions have been found to obey a new multiplication theorem. The cost for correcting this instability is the replacement of a finite series of MacDonald functions by an infinite series via the multiplication theorem for the full MacDonald function.