Impact-Parameter Method for Proton—Hydrogen-Atom Collisions. I. The Time-Dependent Impact-Parameter Model

The impact-parameter method used to calculate cross sections in high-energy proton—hydrogen-atom collisions is considered. By making use of the observation that the scattering is almost completely in the forward direction, a complete time-dependent impact-parameter model is obtained from the Lippmann-Schwinger equation. As a direct consequence of the proper treatment of the continuum, two equivalent Schrödinger time representations are found in which to describe the model system of a point charge moving along a specified trajectory, and thereby perturbing a hydrogen atom. The unambiguous identification of the dynamical states of this model system, and the unambiguous definition of transition probabilities are obtained. The specification of the dynamical states of the time-dependent model system can be conveniently incorporated into the customary impact-parameter method by associating an additional boundary condition with the usual time-dependent Schrödinger equation. The familiar traveling atomic orbitals are not dynamical states of the system, but the discrete dynamical states approach the traveling orbitals as $\mathrm{}\left|t\right|\mathrm{}\to \infty$; and hence, the set of traveling atomic orbitals provides a description of the system which becomes correct asymptotically. One Schrödinger representation is the natural representation in which to obtain the amplitudes for electronic excitation. The other representation is the natural representation in which to obtain the amplitudes for charge exchange. A noniterative technique is used to solve the integral equations which describe the evolution of the state vector in the two time representations, and expressions are derived for the amplitudes for excitation and charge exchange. The amplitudes for excitation can be expressed in terms of the usual Coulomb integrals, and the amplitudes for charge exchange can be expressed in terms of usual exchange integrals.