First-Order Amplitude for General State-to-State Transitions in Hydrogen by Projectile Impact

摘要

The closed analytic form for bound-state transitions due to projectile impact is found in the intermediate representation. The coordinate integral is obtained by evaluating the remaining two integrals in the general multicenter integral derived previously [J.C. Straton, Phys. Rev. A 41, 71 (1990)]. Evaluating the remaining time integral depends upon relating a sum of modified Bessel functions of the second kind KN+1/2(z) to a simple polynomial in 1/z. The results of Van Den Bos and De Heer [Physica 34, 333 (1967)] are shown to be missing a phase factor of (-i)(l′+l)