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Physical Review A

Bohm’s quantum-force time series: Stable distribution, flat power spectrum, and implication

Boon Leong Lan

Published Mar 16, 2001• 13 citations
Publisher

Abstract

New insight into the nature of Bohm's quantum force is presented. Based on our numerical study, we conclude that, for the kicked pendulum, Bohm's quantum-force time-series for nonstationary states is typically or generically non-Gaussian stable distributed with a flat power spectrum. For fixed system parameters and initial wave function, the stable parameters and the constant value of the power spectrum are independent of the initial Bohmian angle. We conjecture that these properties of the quantum-force time-series are also typical or generic for other classically chaotic Hamiltonian dynamical systems since the kicked pendulum is a prototypical member of this class of systems. A new method of calculating the quantum probability density of a particle's position implied by these quantum-force properties is described.

DOI
10.1103/physreva.63.042105

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New insight into the nature of Bohm's quantum force is presented. Based on our numerical study, we conclude that, for the kicked pendulum, Bohm's quantum-force time-series for nonstationary states is typically or generically non-Gaussian stable distributed with a flat power spectrum. For fixed system parameters and initial wave function, the stable parameters and the constant value of the power spectrum are independent of the initial Bohmian angle. We conjecture that these properties of the quantum-force time-series are also typical or generic for other classically chaotic Hamiltonian dynamical systems since the kicked pendulum is a prototypical member of this class of systems. A new method of calculating the quantum probability density of a particle's position implied by these quantum-force properties is described.
The nature of the stochastic process defined by Bohm's momentum is elucidated for stationary energy eigenstates and nonstationary states in classically nonchaotic and chaotic Hamiltonian systems. In addition, the nature of the stochastic process defined by Bohm's quantum force is elucidated for stationary energy eigenstates in nonchaotic and chaotic systems, and for nonstationary states in nonchaotic systems. From these results, the following can be concluded. For stationary energy eigenstates, the process defined by the momentum is generically a stationary, Dirac-delta process in both nonchaotic and chaotic systems; in contrast, the process defined by the quantum force is nongeneric. For nonstationary states, the processes defined by the momentum and quantum force are both nongeneric in nonchaotic systems. Furthermore, for nonstationary states, the process defined by the momentum is, with a high level of confidence, a stationary, stable, brown ${(f}^{\ensuremath{-}2}$ power spectrum) process in the chaotic kicked pendulum. It is conjectured that this is also true for other chaotic systems. The preceding conclusion and conjecture complement those in [Phys. Rev. A 63, 042105 (2001)] for the process defined by the quantum force for nonstationary states in chaotic systems.
The behavior of the periodically kicked pendulum is determined classically and quantum mechanically. Our earlier analytic study [Phys. Rev. A 41, 2952 (1990)] is confirmed and extended by the numerical results presented here. We are able to construct minimum uncertainty, Gaussian wave packets that maintain their form, and shadow the corresponding classical motion for long times when the classical motion is nonchaotic or only weakly chaotic. However, when the classical motion is robustly chaotic, the quantum-mechanical probability distribution grows macroscopically broad exponentially fast and the quantum-classical correspondence breaks down.
The trajectory predicted by Bohm's quantum equation of motion, using an initially angle-localized minimum-uncertainty Gaussian wave packet, is compared with the trajectory predicted by Newton's classical equation of motion for a macroscopic periodically kicked pendulum system. We find that, in violation of the strong form of the correspondence principle, the Bohmian trajectory does not, in general, agree or correspond with the Newtonian trajectory for all times. The Bohm-Newton trajectory correspondence holds as long as the wave packet remains localized in angle. Delocalization of the wave packet in angle triggers the breakdown of the correspondence. The breakdown occurs exponentially quickly if either the Newtonian trajectory that initially coincides with the Bohmian trajectory or one of its classical neighbors is chaotic. In contrast, the breakdown occurs much slower if neither the Newtonian trajectory nor its neighbors are chaotic. These behaviors of the Bohm-Newton trajectory correspondence are generic, or universal, for the class of macroscopic Hamiltonian dynamical systems with bound coordinate space since the kick pendulum is a prototypical member. We predict that neither Bohm's equation of motion nor Newton's correctly predicts the experimentally observed and measured trajectory of a macroscopic system after the breakdown of the correspondence, implying that the all-times agreement required by the strong form of the correspondence principle is unnecessary.
We consider a two-state system kicked quasiperiodically by an external force. When the two kicking frequencies assumed for the force are incommensurate, there can be quantum chaos in the sense that (a) the autocorrelation function of the state vector decays, (b) the power spectrum of the state vector is broadband, and (c) the motion on the Bloch sphere is ergodic. The time evolution of the state vector is nevertheless dynamically stable in the sense that memory of the initial state is retained. We also consider briefly the kicked quantum rotator and find, in agreement with Shepelyansky [Physica 8D, 208 (1983)], that the quantum localization effect is greatly weakened by the presence of two incommensurate driving frequencies.
This paper presents a number of numerical investigations of orbits in the de Broglie–Bohm version of quantum mechanics. We first clarify how the notion of chaos should be implemented in the case of Bohmian orbits. Then, we investigate the Bohmian orbits in three different characteristic quantum systems: (a) superposition of three stationary states in the Hamiltonian of two uncoupled harmonic oscillators with incommensurable frequencies, (b) wave packets in a Hénon–Heiles-type Hamiltonian and (c) a modified two-slit experiment. In these examples, we identify regular or chaotic orbits and also orbits exhibiting a temporarily regular and then chaotic behaviour. Then, we focus on a numerical investigation of the Bohm–Vigier (Bohm and Vigier 1954 Phys. Rev. 26 208) theory, that an arbitrary initial particle distribution P should asymptotically tend to |ψ| 2 , by considering the role of chaotic mixing in causing irregularity of Madelung's flow, a necessary condition for P to tend to |ψ| 2 . We find that the degree of chaos of a particular system correlates with the speed of convergence of P to |ψ| 2 . In the case of wave-packet dynamics, our numerical data show that the time of convergence scales exponentially with the inverse of the effective perturbation from the harmonic oscillator Hamiltonian. The latter result can be viewed as a quantum analogue of Nekhoroshev's (Nekhoroshev 1977 Russ. Math. Surveys 32 1) theorem of exponential stability in classical nonlinear Hamiltonian dynamics.