A trajectory method has been developed for the solution of the one-electron Schrodinger equation in atomic scattering problems. The method is based on the fluid dynamical interpretation of the quantum mechanical probability density and current. The time evolution is described by trajectories of particles, which represent the probability distribution. The trajectories are calculated by solving classical equations of motion with forces depending on the probability density. Results are obtained for the differential and total ionisation and charge exchange cross sections for the scattering of H + on H.
Based on Bohmian quantum trajectory theory, an investigation of electron scattering in a thin crystal has been performed. A time-independent interaction potential was employed to describe electron scattering from model crystal. Quantum trajectories were calculated by a numerical solution of the time-dependent Schrödinger equation. The probability density functions and quantum trajectories representing the electron diffraction in real space were obtained from the calculation. The quantum trajectories provide an intuitive dynamics of the interaction process during the electron diffraction.
Time-independent wave-packet forms of the Schr\"odinger equation (TIWSE) and Lippmann-Schwinger equation (TIWLSB) have been derived by a partial time-to-energy Fourier transform of
L2 wave-packet solutions to the time-dependent Schr\"odinger equation. The new equations retain the initial wave packet \ensuremath{\chi}(
t0) as a ``universal source'' of scattered waves, which applies for all collision energies E contained in the initial wave packet. The relationship between the solution ${\mathrm{\ensuremath{\Psi}}}_{\mathit{t}0}$(E) of the TIWSE or TIWLSE and the scattering solution ${\mathrm{\ensuremath{\Psi}}}^{(+)}$(E) of the standard time-independent Lippmann-Schwinger or Schr\"odinger equation is given and the method illustrated by a computational application.