General state-to-state transitions in atoms for either Yukawa or Coulomb potentials via Fourier transforms

Abstract

In a previous paper we used our integro-differential extension of Gaussian transforms to find the closed analytic form for hydrogenic bound-state transitions with arbitrarily high quantum numbers due to projectile impact excitation via Coulomb potentials (in the intermediate representation). Here we extend that result to Yukawa potentials, but do so by utilizing Fourier transforms. The result is used to find the first-order cross section for proton impact excitation of hydrogen to the 2s–7s final states. Because the results hold for any initial and final quantum states, and the amplitude may be easily converted for use with pseudostates, it may be used to automatically calculate sums over intermediate pseudostate propagators whose (bound and free) principal and angular quantum numbers can become very large. This result may be extended to multi-electron transition amplitudes by representing initial and final states by configuration–interaction wave-functions.