Finite-sample deviations and convergence in the statistics of Bohmian trajectory ensembles

TL;DR

This paper investigates how finite sample sizes affect the statistical properties of Bohmian trajectories, revealing conditions under which they align with quantum predictions and highlighting complexities in chaotic and superposed systems.

Contribution

It provides a detailed analysis of finite-sample deviations in Bohmian trajectory ensembles, especially in complex and chaotic quantum systems, and offers guidance for numerical simulations.

Findings

Sample means and variances match Born-rule moments in regular flows.

Chaotic and superposed systems show sensitive dependence and deviations.

Different velocity fields in spin-1/2 particles can lead to varied finite-sample statistics.

Abstract

We analyze finite-sample statistics of Bohmian trajectories for single spinless and spin-1/2 particles. Equivariance ensures agreement with $|\psi|^2$ in the quantum equilibrium limit, yet experiments and simulations necessarily use finite ensembles. We show that in regular flows (e.g., wavepackets or low-mode superpositions of eigenstates of harmonic oscillators) sample means and/or variances over modest $N$ are consistent with Born-rule moments. In contrast, degenerate superpositions of 3D oscillators with nodal barriers and chaotic Bohmian dynamics exhibit sensitive dependence on initial conditions and complex flow partitioning, which can yield noticeable finite-sample deviations in the mean and variance. For the spin-1/2 particle, both convective and Pauli currents conserve $|\psi|^2$, but they are associated with different velocity fields and thus might yield different finite-sample trajectory statistics. These findings calibrate the interpretation of trajectory-based uncertainty and provide practical guidance for numerical Bohmian simulations of spin and transport, without challenging the equivalence to orthodox quantum mechanics in the quantum equilibrium ensemble.