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Journal of Physics B Atomic and Molecular Physics

Trajectory method for the solution of the time-dependent Schrodinger equation in atomic physics and application to H+-H scattering

Georg Terlecki, NORBERT GRÜN, W. Scheid

Published Sep 28, 1984• 20 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/17/18/011

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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.
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A connection between quantum mechanics and classical methods used in ion-atom collisions is developed. The connection is based on the fluid dynamical interpretation of the one-electron quantum wavefunction. The classical trajectory Monte Carlo method and the Thomson approximation are obtained as quantum models and the wavefunctions associated with these approximations are defined. A discussion is given of the procedure to be used to calculate scattering cross sections. Scaling laws derivable from the classical Liouville equation are deduced in the impact parameter formalism.
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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.
Transcending its origin in the debate on the interpretation of quantum theory, the continuous deterministic trajectory has proved a potent tool in computational quantum mechanics (an excellent introduction to numerical trajectory techniques is provided by Wyatt’s text [1]). It is straightforward to develop a method to derive the timedependent Schrödinger equation from a single-valued continuum of spacetime trajectories, and supply a corresponding exact formula for the wavefunction [2].Anatural language for this theory of evolution is offered by the hydrodynamic analogy, in which wave mechanics corresponds to the Eulerian picture (as shown by Madelung [3]) and the trajectory theory to the Lagrangian picture. The Lagrangian model for the quantum fluid may be developed from a variational principle involving a specific interaction potential (the quantum internal potential energy), and the Euler-Lagrange equations imply a nonlinear partial differential equation to calculate the trajectories of the fluid particles as functions of their initial coordinates and the initial wavefunction. The latter supplies two sets of data to the particle model: the initial density (via the amplitude) and the initial velocity (via the phase). The time-dependent wavefunction is then computed via the standard map between the Lagrangian coordinates and the Eulerian fields.