Quantum Dynamics via Score Matching on Bohmian Trajectories

2026-04-27

AI Summary

This work addresses the computational bottleneck in solving high-dimensional time-dependent Schrödinger equations by proposing a Bohmian-trajectory-based, score-driven self-consistent continuous normalizing flow method. The approach parameterizes the gradient of the log-probability density (the score function) with neural networks and evolves the quantum dynamics by minimizing a self-consistent Fisher divergence. It reframes real-time quantum dynamics as a self-consistent score-matching problem—a formulation established here for the first time—and theoretically demonstrates that zero-loss solutions exactly reproduce the Schrödinger evolution of nodeless wavefunctions. The method achieves high accuracy and favorable scalability, successfully simulating wavepacket splitting in a double-well potential and anharmonic vibrations in a Morse chain within realistic quantum systems.

Abstract

We solve the time-dependent Schrödinger equation by learning the score function, the gradient of the log-probability density, on Bohmian trajectories. In Bohm's formulation of quantum mechanics, particles follow deterministic paths under the classical potential supplemented by a quantum potential depending on the score function of the evolving density. These non-crossing Bohmian trajectories form a continuous normalizing flow governed by the score. We parametrize the score with a neural network and minimize a self-consistent Fisher divergence between the network and the score of the resulting density. We prove that the zero-loss minimizer of this self-consistent objective recovers Schrödinger dynamics for nodeless wave functions, a condition naturally met in quantum vibrations of atoms. We demonstrate the approach on wavepacket splitting in a double-well potential and anharmonic vibrations of a Morse chain. By recasting real-time quantum dynamics as a self-consistent score-driven normalizing flow, this framework opens the time-dependent Schrödinger equation to the rapidly advancing toolkit of modern generative modeling.

Problem

• time-dependent Schrödinger equation
• quantum dynamics
• Bohmian trajectories
• score function
• wave function evolution

Innovation

• score matching
• Bohmian trajectories
• quantum dynamics
• normalizing flow
• neural quantum states