[Paper Review] Quantum Mechanics with Trajectories: Quantum Trajectories and Adaptive Grids

Róbert E. Wyatt, Eric R. Bittner|ArXiv.org|Feb 11, 2003

Quantum Mechanics and Applications13 references3 citations

TL;DR

This paper presents a computational framework for solving quantum dynamics using quantum trajectories within a hydrodynamic formulation, employing adaptive grids to maintain accuracy during wavepacket scattering and tunneling. By dynamically adjusting grid resolution based on wavefunction curvature and quantum potential gradients, the method enables stable, long-time propagation of quantum systems, particularly in classically forbidden regions and near caustics.

ABSTRACT

Although the foundations of the hydrodynamical formulation of quantum mechanics were laid over 50 years ago, it has only been within the past few years that viable computational implementations have been developed. One approach to solving the hydrodynamic equations uses quantum trajectories as the computational tool. The trajectory equations of motion are described and methods for implementation are discussed, including fitting of the fields to gaussian clusters.

Motivation & Objective

• Address the instability and numerical breakdown in quantum trajectory methods when wavefunction nodes or caustics form during propagation.
• Develop a robust computational approach to simulate quantum dynamics in multidimensional systems using hydrodynamic formulations.
• Overcome limitations of fixed grids by introducing adaptive mesh refinement that follows evolving probability fluid density and curvature.
• Enable accurate computation of the quantum potential and non-local effects in systems with complex dynamics such as barrier tunneling and wavepacket bifurcation.
• Facilitate long-time simulations of quantum systems by dynamically adjusting grid resolution to concentrate computational resources where needed most.

Proposed method

• Formulate quantum dynamics using the hydrodynamic (Bohmian) approach, where the wavefunction is decomposed into amplitude and phase to derive equations of motion for fluid elements.
• Implement a Lagrangian, moving-with-the-fluid framework to track the evolution of quantum trajectories and hydrodynamic fields.
• Use a monitor function based on local wavefunction curvature to guide adaptive grid refinement, ensuring high resolution in regions of high probability density and rapid changes.
• Apply the Dorfi and Drury adaptive grid algorithm to control grid point clustering and prevent numerical instabilities from sudden grid distortions.
• Compute the quantum potential Q on-the-fly from the wavefunction's amplitude and phase, incorporating non-classical effects such as tunneling and interference.
• Employ a spring-like analogy (smart springs) to regulate grid motion and maintain smooth, stable grid evolution during propagation.

Experimental results

Research questions

• RQ1 How can quantum trajectory methods be stabilized during long-time propagation when wavefunction nodes or caustics develop?
• RQ2 What adaptive grid strategy best maintains accuracy and resolution in regions of high quantum potential and wavefunction curvature?
• RQ3 Can the hydrodynamic formulation with adaptive grids accurately simulate wavepacket scattering and tunneling through repulsive barriers?
• RQ4 How do dynamic grid adjustments affect the stability and computational efficiency of quantum trajectory simulations?
• RQ5 What are the limitations of current adaptive grid techniques in handling stiff hydrodynamic equations near classical turning points and nodes?

Key findings

• Adaptive grids enabled stable propagation of a Gaussian wavepacket through a repulsive Eckart barrier beyond the point where fixed-grid methods failed due to node formation.
• Wavepacket bifurcation into transmitted and reflected components was clearly resolved by grid clustering in the respective regions, with grid paths diverging at around 40 fs.
• The grid dynamically concentrated points in high-curvature and high-density regions, such as near the barrier and in the transmitted/reflected wavepackets, improving resolution where needed.
• The use of a monitor function based on wavefunction curvature effectively guided grid adaptation, preventing grid collapse and maintaining numerical stability.
• The method successfully captured non-classical effects such as tunneling and interference, with the quantum potential computed on-the-fly to include all quantum features.
• Long-time simulations were achieved by avoiding fixed grids and instead using a moving, adaptive mesh that followed the evolving probability fluid, reducing computational waste in low-activity regions.