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arXiv:quant-ph/0701190 · The Journal of Physical Chemistry A

Quantum Dynamics with Bohmian Trajectories

Dirk-André Deckert, D. Dürr, Peter Pickl

Published Jul 13, 2007• 19 citations
arXivHTMLPDF

Abstract

We describe the advantages and disadvantages of numerical methods when Bohmian trajectory grids are used for numerical simulations of quantum dynamics. We focus on the crucial noncrossing property of Bohmian trajectories, which, numerically, must be given careful attention. Failure to do so causes instabilities or leads to false simulations.

DOI
10.1021/jp0711996

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We describe the advantages and disadvantages of numerical methods when Bohmian trajectory grids are used for numerical simulations of quantum dynamics. We focus on the crucial noncrossing property of Bohmian trajectories, which, numerically, must be given careful attention. Failure to do so causes instabilities or leads to false simulations.
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In this work we present a numerical analysis of the so-called ‘interacting quantum trajectory method’ (IQTM), which is based on a formal theory developed by Poirier [Bohmian mechanics without pilot waves, Chem. Phys. 370 (2010), pp. 4–14.], and later extended by Schiff and Poirier [Communication: Quantum mechanics without wavefunctions, J. Chem. Phys. 136 (2012), p. 031102.], for representing the quantum mechanical state of a particle in terms of an ensemble of Bohmian quantum trajectories. This has led to a number of numerical algorithms over the years, mostly for time-independent applications. A numerically robust, exact time-dependent wavepacket propagation scheme would have to await the refinements of Dupuy et al. [L. Dupuy, F. Talotta, F. Agostini, D. Lauvergnat, B. Poirier, and Y. Scribano, Adiabatic and nonadiabatic dynamics with interacting quantum trajectories, J. Chem. Theory Comput. 18 (2022), pp. 6447–6462.], which serves as the starting point for our continued development in the present study. Here we will discuss the numerical protocols comprising IQTM, and present detailed error analysis for several 1D model problems involving simple free particles and oscillators. These studies reveal some key advantages of IQTM; however, they also highlight problems for which the algorithms need further work.
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