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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Numerical analysis of the interacting quantum trajectory method

Jeremy B. Maddox, Mikhail Khenner, Bill Poirier

Published Sep 1, 2026• 0 citations

Abstract

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.

DOI: 10.1080/14786435.2026.2726234

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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.

We present a quantum dynamics method based on the propagation of interacting quantum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [ Chem. Phys. 2010, 370, 4–14] and Schiff and Poirier [ J. Chem. Phys. 2012, 136, 031102] on quantum dynamics without wavefunctions. It consists of determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quantum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

We present a quantum dynamics method based on the propagation of interacting quan- tum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys. 370 4–14 (2010)] and Schiff and Poirier [J. Chem. Phys. 136 031102 (2012)] on quantum dynamics with- out wavefunctions. It consists in determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quan- tum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

We present a quantum dynamics method based on the propagation of interacting quan- tum trajectories to describe both adiabatic and nonadiabatic processes within the same formalism. The idea originates from the work of Poirier [Chem. Phys. 370 4–14 (2010)] and Schiff and Poirier [J. Chem. Phys. 136 031102 (2012)] on quantum dynamics with- out wavefunctions. It consists in determining the quantum force arising in the Bohmian hydrodynamic formulation of quantum dynamics using only information about quan- tum trajectories. The particular time-dependent propagation scheme proposed here results in very stable dynamics. Its performance is discussed by applying the method to analytical potentials in the adiabatic regime, and by combining it with the exact factorization method in the nonadiabatic regime.

Quantum trajectory methods (QTMs) hold great promise as a potential means of obtaining dynamical insight and computational scaling similar to classical trajectory simulations but in an exact quantum dynamical context. To date, the development of QTMs has been stymied by the "node problem"--highly nonclassical and numerically unstable trajectories that arise when the wavepacket density |psi|2 exhibits substantial interference oscillations. In a recent paper, however [B. Poirier, J. Chem. Phys. 128, 164115 (2008)], a "bipolar decomposition," psi=psi+(+)(psi)psi(-), was introduced for one-dimensional (1D) wavepacket dynamics calculations such that the component densities |psi(+)(-)|2 are slowly varying and otherwise interference-free, even when |psi|2 itself is highly oscillatory. The bipolar approach is thus ideally suited to a QTM implementation, as is demonstrated explicitly in this paper. Two model 1D benchmark systems exhibiting substantial interference are considered--one with more "quantum" system parameters and the other more classical-like. For the latter, more challenging application, synthetic QTM results are obtained and found to be extremely accurate, as compared to a corresponding fixed-grid calculation. Ramifications of the bipolar QTM approach for the classical limit and also for multidimensional applications, are discussed.

In recent decades, the de Broglie-Bohm or ‘pilot wave’ interpretation of quantum mechanics–especially its concept of the ‘quantum trajectory’–has inspired a plethora of computational methods in molecular and chemical physics. One particularly promising subclass of such methods are the ‘interacting quantum trajectory’ (IQT) methods, for which an ensemble of trajectories is propagated independently of–and without any recourse to–an underlying wavefunction. In this work, an IQT version of the Pauli equation is developed–enabling spin to be incorporated into IQT theories for the first time, to the authors' knowledge. The new trajectory-based dynamical equations are then used to solve two iconic quantum problems involving spin: the ‘quantum spin flipper’, and the Stern-Gerlach experiment.

In previous articles [B. Poirier J. Chem. Phys. 121, 4501 (2004); C. Trahan and B. Poirier, ibid. 124, 034115 (2006); 124, 034116 (2006); B. Poirier and G. Parlant, J. Phys. Chem. A 111, 10400 (2007)] a bipolar counterpropagating wave decomposition, psi = psi(+) + psi(-), was presented for stationary states psi of the one-dimensional Schrodinger equation, such that the components psi(+/-) approach their semiclassical Wentzel-Kramers-Brillouin analogs in the large action limit. The corresponding bipolar quantum trajectories are classical-like and well behaved, even when psi has many nodes, or is wildly oscillatory. In this paper, the method is generalized for time-dependent wavepacket dynamics applications and applied to several benchmark problems, including multisurface systems with nonadiabatic coupling.