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

Abstract

We analyze finite-sample statistics of Bohmian trajectories for single spinless and spin-1/2 particles. Equivariance ensures agreement with $|ψ|^2$ 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 $|ψ|^2$, 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.

Figures (11)

• Figure 1: The motion of a quantum harmonic oscillator with mass $M$ and frequency $\omega$, released from a superposition of a few eigenstates. The initial state in panels (a) and (b) is given by Eq. \ref{['eq:harsup1']}; in panels (c) and (d), by Eq. \ref{['eq:harsup2']}. Panels (a) and (c) indicate the mean position, while (b) and (d) present the standard deviation of position, as functions of time. In all panels, the black solid lines indicate the results of statistical analysis based on 400 trajectories, whereas the red dashed lines correspond to calculations employing the probabilistic character of the wavefunction. Time and displacement are measured in units of $\omega^{-1}$ and $(\hbar/M\omega)^{-1/2}$, respectively.
• Figure 2: The motion of a quantum harmonic oscillator with mass $M$ and frequency $\omega$, released from a wavepacket centered at $x_c=1$ with the spread $\gamma^2=0.5$ and zero phase, c.f. Eq. \ref{['eq:harmpt']}. Panels (a) and (b) show the time dependence of the mean and the standard deviation of position, respectively. In all panels, the black solid lines indicate the results of statistical analysis based on 400 trajectories, whereas the red dashed lines correspond to calculations employing the probabilistic character of the wavefunction. Time and displacement are measured in units of $\omega^{-1}$ and $(\hbar/M\omega)^{-1/2}$, respectively.
• Figure 3: The motion of a free particle with a unit mass $M$, released from a wavepacket centered at $x_c=0$ with spread $\gamma^2=0.5$ and zero phase. Panels (a) and (b) show the time dependence of the mean and standard deviation of position, respectively. In all panels, the black solid lines indicate the results of statistical analysis based on 400 trajectories, whereas the red dashed lines correspond to calculations employing the probabilistic character of the wavefunction. In panel (b), black and red lines overlap. The Planck constant is taken to be $\hbar=1$ .
• Figure 4: The motion of a 3D quantum harmonic oscillator with mass $M$ and oscillatory frequency $\omega$, released from an initial state in Eq. \ref{['eq:harm3D']}. Panels (a) and (b) show the time dependence of the mean and standard deviation of position, respectively. In all panels, the black solid lines indicate the results of statistical analysis based on 250 trajectories, whereas the red dashed lines correspond to calculations employing the probabilistic character of the wavefunction. Time and displacement are measured in units of $\omega^{-1}$ and $(\hbar/M\omega)^{-1/2}$, respectively.
• Figure 5: The motion of a spin-1/2 neutral particle with a unit mass subjected to external time-dependent magnetic field $\mathbf{B}=(0,-by,bz)$, released from a wavepacket centered at the origin with spread $\gamma^2=100$ in all directions, whose momentum of the $y$-component of the wavepacket is $0.08$ while other components are zero. The spin part is aligned with the -$z$-axis. Panels (a) and (b) show the time dependence of the mean position along the $x$- and $z$-axes, respectively, where the starting points of all curves are shifted to the origin. In all panels, the black and orange solid lines are the statistical analysis based on 400 trajectories of convective and Pauli currents, respectively. Calculations employing the probabilistic character of the wavefunction are represented by red dashed lines. The magnetic field $b$ is chosen to be 0.1 and the magnetic moment is $\mu=-0.001$. Atomic units are used in the calculation unless otherwise specified.