Polarization in distant Coulomb collisions of charged particles with atoms

The inelastic collisions of a heavy fast particle of charge $Z_{1}e$, incident at an impact parameter $b$, with an electron bound in an atom isotropically and harmonically is considered. The "atom" is treated by quantum mechanics, extending an earlier classical calculation by Ashley, Ritchie, and Brandt. A multiople of the Coulomb interaction for distanct collisions is used. Two methods are employed to calculate $Z_1^3$ contributions to the mean energy loss and the excitation probabilities due to dipole and quadrupole terms in the multipole expansion. In one method the dipole interaction is taken into account exactly by applying the quantum formalism of the forced harmonic oscillator, and the quadrupole interaction is treated as first-order perturbation. The second method of calculation employs both dipole and quadrupole interactions as time-dependent perturbations to second order on the free harmonic oscillator. The energy loss calculated for a particle incident on a straight-line trajectory agrees with the classical calculation. Reasons for this agreement are goven. Simplifications of the calculation owing to time-reversal symmetry considerations and selection rules for the harmonic oscillator are discussed. Comments are made concerning the relative importance of distant and close collisions at high and low velocities.