Stable direct dynamics with quantum potential: Lorentzian trajectory basis function is all you need

Abstract

Quantum trajectory methods—such as those based on the de Broglie–Bohm and multiple-interacting-worlds formulations—offer conceptually appealing alternatives to conventional wavefunction-based quantum mechanics. A persistent challenge in their practical implementation is the instability of the quantum potential, particularly in low-density regions. This work introduces a strategy to enable stable, robust trajectory-based quantum dynamics by constructing the probability density via superposition of Lorentzian-shaped trajectory basis functions (TBFs). Compared to commonly used Gaussian TBFs, Lorentzian functions feature a cusp at the origin and slower asymptotic decay, resulting in bounded and smooth quantum potentials and well-behaved quantum forces. A general principle is proposed for selecting TBFs suitable for constructing quantum potentials in trajectory-based simulations. These findings offer a promising direction for improving the stability of quantum trajectory integration and may benefit a wide class of coupled-trajectory and quantum–classical methods. The proposed TBFs also merit consideration across diverse quantum simulation domains, including wavepacket propagation and electronic structure theory.