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arXiv:1008.2025

The reproduction of the dynamics of a quantum system by an ensemble of classical particles beyond de Broglie--Bohmian mechanics

Denys I. Bondar

Published Aug 11, 2010• 0 citations
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Abstract

It is shown that for any given quantum system evolving unitarily with the Hamiltonian, $\hat{H} = \hat{\bf p}^2/(2m) + U({\bf q})$, [bold letters denote $D$-dimensional ($D \geqslant 3$) vectors] and with a sufficiently smooth potential $U({\bf q})$, there exits a classical ensemble with the Hamilton function, $\mathrsfs{H} ({\bf p}, {\bf q}) = {\bf p}^2/(2m) + U^{(\infty)} ({\bf q})$, where the potential $U^{(\infty)}({\bf q})$ coincides with $U({\bf q})$ for almost all ${\bf q}$ (i.e., $U^{(\infty)}$ can be different from $U$ only on a measure zero set), such that the square modulus of the wave function in the coordinate (momentum) representation approximately equals the coordinate (momentum) distribution of the classical ensemble within an arbitrary given accuracy. Furthermore, the trajectories of this classical ensemble, generally speaking, need not coincide with the trajectories obtained from de Broglie--Bohmian mechanics. Consequences of this result are discussed.

DOI
10.48550/arxiv.1008.2025

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It is shown that for any given quantum system evolving unitarily with the Hamiltonian, H^=p^2/(2m)+U(q)\hat{H} = \hat{\bf p}^2/(2m) + U({\bf q})H^=p^​2/(2m)+U(q), [bold letters denote DDD-dimensional (D⩾3D \geqslant 3D⩾3) vectors] and with a sufficiently smooth potential U(q)U({\bf q})U(q), there exits a classical ensemble with the Hamilton function, \mathrsfsH(p,q)=p2/(2m)+U(∞)(q)\mathrsfs{H} ({\bf p}, {\bf q}) = {\bf p}^2/(2m) + U^{(\infty)} ({\bf q})\mathrsfsH(p,q)=p2/(2m)+U(∞)(q), where the potential U(∞)(q)U^{(\infty)}({\bf q})U(∞)(q) coincides with U(q)U({\bf q})U(q) for almost all q{\bf q}q (i.e., U(∞)U^{(\infty)}U(∞) can be different from UUU only on a measure zero set), such that the square modulus of the wave function in the coordinate (momentum) representation approximately equals the coordinate (momentum) distribution of the classical ensemble within an arbitrary given accuracy. Furthermore, the trajectories of this classical ensemble, generally speaking, need not coincide with the trajectories obtained from de Broglie--Bohmian mechanics. Consequences of this result are discussed.
Bohmian mechanics is endowed with an ontological package that supposedly allows to solve the main interpretational problems of quantum mechanics. We are concerned in this work by the emergence of classicality from the quantum mechanical substrate. We will argue that although being superficially attractive, the de Broglie-Bohm interpretation does not shed new light on the quantum-to-classical transition. This is due to nature of the dynamical law of Bohmian mechanics by which the particles follow the streamlines of the probability flow. As a consequence, Bohmian trajectories can be highly non-classical even when the wavefunction propagates along classical trajectories, as happens in semiclassical systems. In order to account for classical dynamics, Bohmian mechanics needs non-spreading and non-interfering wave packets: this is achieved for practical purposes by having recourse to decoherence and dense measurements. However one then faces the usual fundamental problems associated with the meaning of reduced density matrices. Moreover the specific assets of the de Broglie-Bohm interpretation – in particular the existence of point-like particles pursuing well-defined trajectories – would play no role in accounting for the emergence of classical dynamics.
Abstract Bohmian mechanics supplements the quantum wavefunction with deterministic particle trajectories, offering an alternate, dynamical language for quantum theory. However, the Bohmian wavefunction evolves independently of these trajectories, and is thus unaffected by the observable properties of the system. While this property is widely assumed necessary to ensure agreement with quantum mechanics, much work has recently been dedicated to understanding classical pilot-wave systems, which feature a two-way coupling between particle and wave. These systems—including the “walking droplet” system of Couder and Fort (Couder and Fort (2006) Phys. Rev. Lett. 97:154101) and its various abstractions (Dagan and Bush (2020) CR Mecanique 348:555–571; Durey and Bush (2020) Front. Phys. 8:300; (2021) Chaos 31:033136; Darrow and Bush (2024) Symmetry 16:149)—allow us to investigate the limits of classical systems and offer a touchstone between quantum and classical dynamics. In this work, we present a general result that bridges Bohmian mechanics with this classical pilot-wave theory. Namely, Darrow and Bush ((2024) Symmetry 16:149) recently introduced a Lagrangian pilot-wave framework to study quantum-like behaviours in classical systems; with a particular choice of particle-wave coupling, they recover key dynamics hypothesised in de Broglie’s early double-solution theory (de Broglie (1970) Foundations Phys. 1:5–15). We here show that, with a different choice of coupling, their de Broglie-like system reduces exactly to single-particle Bohmian mechanics in the non-relativistic limit. Our result clarifies that, while multi-particle entanglement is impossible to replicate in general with local, classical theories, no such restriction exists for single-particle quantum mechanics. Moreover, connecting with the previous work of Darrow and Bush, our work demonstrates that de Broglie’s and Bohm’s theories can be connected naturally within a single Lagrangian framework. Finally, we present an application of the present work in developing a single-particle analogue for position measurement in a de Broglie-like setting.
de Broglie-Bohm theory (dBBT), treating quantum particles as point objects moving along well defined (Bohmian) trajectories, offers an appealing solution of the measurement problem in quantum mechanics; it has, however, problems relating to spin, relativity and lack of proper integration with the Hilbert space based framework. In this work, we present a consistent formalism which has the traditional state-observable framework integrated with the desirable features of dBBT. We adopt ensemble interpretation for the Schrodinger wave function ψψψ. Given a Schrodinger wave function ψψψ, we use its value ψ0ψ_0ψ0​ at some fixed time (say, t=0t = 0t=0) to define the probability measure ∣ψ0∣2dx|ψ_0|^2 {\rm d}x∣ψ0​∣2dx on the system configuration space MMM (=Rn=\mathbb{R}^n=Rn). On the resulting probability space M0\mathcal{M}_0M0​, we introduce a stochastic process ξ(t)ξ(t)ξ(t) corresponding to the Heisenberg position operator XH(t)X_H(t)XH​(t) such that, in the Heisenberg state ∣ψh⟩|ψ_h\rangle∣ψh​⟩ corresponding to ψ0ψ_0ψ0​, the expectation value of XH(t)X_H(t)XH​(t) equals that of ξ(t)ξ(t)ξ(t) in M0\mathcal{M}_0M0​. This condition leads to the de Broglie-Bohm guidance equation for the sample paths of the process ξ(t)ξ(t)ξ(t) which are, therefore, Bohmian trajectories supposedly representing time-evolutions of individual members of the ψ0ψ_0ψ0​-ensemble. Stochastic processes and Bohmian trajectories corresponding to observables with discrete eigenvalues (in particular spin) are treated by extending the configuration space to the spectral space of the commutative algebra obtained by adding appropriate discrete observables to the position observables. Pauli's equation is treated as an example. A straightforward derivation of von Neumann's projection rule employing the Schrodinger-Bohm evolution of individual systems along their Bohmian trajectories is given. Some comments on the potential application of the formalism developed here to quantum mechanics of the universe are included.
The quantum trajectories in the de Broglie-Bohm formulation of quantum mechanics depend on an additional quantum potential derived from the full wave solution of Schrödinger's equation. The task of supplying collectively all the correct quantum results strongly alters the characteristics of the corresponding classical trajectories, which underlie semiclassical approximations to the evolving wave function. Both classical and quantum trajectories are here considered to be conservative with no influence of an external environment, even though this is the source of eventual classicality in quantum systems, that is, decoherence. The concept of integrability, closely correspondent in classical and quantum mechanics, is not preserved by the quantum trajectories. General systems, in which classical chaotic motion participates, are much harder to treat semiclassically, but quantum trajectories can be chaotic even for integrable systems. This discrepancy between the character of classical and quantum trajectories in the de Broglie-Bohm interpretation does not clarify the singular classical-quantum transition.
Quantum trajectories, originating from the de Broglie–Bohm hydrodynamic description of quantum mechanics, are used to construct time-correlation functions in an initial value representation. The formulation is fully quantum mechanical and the resulting equations for the correlation functions are similar in form to their semiclassical analogs but do not require the computation of the stability or monodromy matrix or conjugate points. We then move to a local trajectory description by evolving the cumulants of the wave function along each individual path. The resulting equations of motion are an infinite hierarchy, which we truncate at a given order. We show that time-correlation functions computed using these approximate quantum trajectories can be used to accurately compute the eigenvalue spectrum for various potential systems.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.