Time dependent formulation of the collision problem in quantum mechanics

Leonard Eisenbud

Jan Linderberg

P. R. Berman

A quantum-mechanical transport equation (QMTE) is derived which should be applicable to a wide range of problems involving the interaction of radiation with atoms or molecules which are also subject to collisions with perturber atoms. The equation follows the time evolution of the macroscopic atomic density matrix elements of atoms located at classical position $\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}$ and moving with classical velocity $\stackrel{\ensuremath{\rightarrow}}{\mathrm{v}}$ It is quantum mechanical in the sense that all collision kernels or rates which appear have been obtained from a quantum-mechanical theory and, as such, properly take into account the energy-level variations and velocity changes of the active (emitting or absorbing) atom produced in collisions with perturber atoms. The QMTE represents a somewhat different formulation of the problem than that considered in earlier works. The present formulation is better suited to problems involving high-intensity external fields, such as those encountered in laser physics.

San-ichiro Mizushima

Kit-Keung Kan, James J. Griffin

The problem of a single quantal particle moving in a time-dependent external potential well is formulated specifically to emphasize and develop the fluid dynamical aspects of the matter flow. This idealized problem, the single-particle Schr\"odinger fluid, is shown to exhibit already a remarkably rich variety of fluid dynamical features, including compressible flow and line vortices. It provides also a sufficient framework to encompass simultaneously various simplified fluidic models for nuclei which have earlier been postulated on an ad hoc basis, and to illuminate their underlying restrictions. Explicit solutions of the single-particle Schr\"odinger fluid problem are studied in the adiabatic limit for their mathematical and physical implications (especially regarding the collective kinetic energy). The basic generalizations for extension of the treatment to the many-body Schr\"odinger fluid are set forth.NUCLEAR STRUCTURE Fluid dynamical formulation of quantum mechanics, single-particle behavior in nuclear collective motion, fission and heavy-ion collision, inertial parameter, adiabatic approximation.

Karl P. Horn, Christiane P. Koch

Collisions between atoms and molecules govern how energy is exchanged and redistributed in the gas phase, underpinning phenomena from spectral line broadening and interstellar molecular excitation to the cooling and trapping of molecules. A quantitative account of these collisions requires connecting the theoretical description of the interaction between the collision partners to measurable quantities such as cross sections. However, scattering theory as presented in quantum mechanics textbooks does not readily translate into numerical methods. Here, we formulate the scattering problem in Dirac notation, as customary in present-day quantum mechanics courses, and show how to build a practical numerical framework from it. We derive the "coupled channel" representation of the scattering problem with correct asymptotic behaviour and introduce Numerov's method to obtain time-independent wavefunctions, from which we calculate cross sections, the observables measured in scattering experiments. Starting from single channel scattering, we build up to the realistic collision of a ground-state atom with a diatomic molecule.