Formulation of the quantum mechanical collision problem

Leonard Eisenbud

San-ichiro Mizushima

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.

Ettabiy Abderrahmane

The fundamental equation of the model titled “mathematical formulation of nuclear collision mechanisms” [1] was solved. The solution is compounded of two steps, the first step consists in calculating the values of the parameters of the mechanism-operator underlying a given nuclear collision and the second step, consists in deducing the open channels for this collision. Knowing that the equation is not classic, its resolution has required the development of an innovative mathematical method that sets on the use of physical constraints of the problem.

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.

William Band, James L Park