A tutorial on quantum scattering theory and coupled-channel calculations of bimolecular collisions
Abstract
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.
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BibTeX
@misc{horn2026tutorial,
title = {{A tutorial on quantum scattering theory and coupled-channel calculations of bimolecular collisions}},
author = {Karl P. Horn and Christiane P. Koch},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20882},
eprint = {2610.20882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20882},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A tutorial on quantum scattering theory and coupled-channel calculations of bimolecular collisions
AU - Karl P. Horn
AU - Christiane P. Koch
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20882
UR - https://arxiv.org/abs/2610.20882
AB - 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.
ER -