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Order in de Broglie- Bohm quantum mechanics

G. Contopoulos, N. Delis, Christos Efthymiopoulos

Published Aug 16, 2016• 18 citations
OpenAlex

Abstract

A usual assumption in the so-called {\it de Broglie - Bohm} approach to quantum dynamics is that the quantum trajectories subject to typical `guiding' wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical systems' sense). In the present paper, we consider mainly cases in which the quantum trajectories are {\it ordered}, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a H\'{e}non - Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie - Bohm formalism, namely the influence of order in the so-called {\it quantum relaxation} effect. We show that the existence of regular trajectories poses restrictions to the quantum relaxation process, and we give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.

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Order in de Broglie–Bohm quantum mechanics

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A usual assumption in the so-called de Broglie–Bohm approach to quantum dynamics is that the quantum trajectories subject to typical 'guiding' wavefunctions turn to be quite irregular, i.e. chaotic (in the dynamical systems' sense). In this paper, we consider mainly cases in which the quantum trajectories are ordered , i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a Hénon–Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie–Bohm formalism, namely the influence of order on the so-called quantum relaxation effect. We show that the existence of regular trajectories poses restrictions on the quantum relaxation process and give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.

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Quantum non-equilibrium and relaxation to equilibrium for a class of de Broglie–Bohm-type theories

Samuel Colin, Ward Struyve

The de Broglie–Bohm theory is about non-relativistic point-particles that move deterministically along trajectories. The theory reproduces the predictions of standard quantum theory, given that the distribution of particles over an ensemble of systems, all described by the same wavefunction ψ, equals the quantum equilibrium distribution |ψ| 2 . Numerical simulations done by Valentini and Westman (2005 Proc. R. Soc. A 461 253) have illustrated that non-equilibrium particle distributions may relax to quantum equilibrium after some time. Here we consider non-equilibrium distributions and their relaxation properties for a particular class of trajectory theories (first studied in detail by Deotto and Ghirardi (1998 Found. Phys. 28 1)) that are empirically equivalent to the de Broglie–Bohm theory in quantum equilibrium. In the examples we studied of such theories, we found a speed-up of the relaxation, compared to the ordinary de Broglie–Bohm theory. Hence non-equilibrium predictions that depend strongly on relaxation properties, such as those studied recently by Valentini, may vary across different trajectory theories. As such, these theories might be experimentally distinguishable.

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A usual assumption in the so-called {\it de Broglie - Bohm} approach to quantum dynamics is that the quantum trajectories subject to typical `guiding' wavefunctions turn to be quite irregular, i.e. {\it chaotic} (in the dynamical systems' sense). In the present paper, we consider mainly cases in which the quantum trajectories are {\it ordered}, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a H'{e}non - Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie - Bohm formalism, namely the influence of order in the so-called {\it quantum relaxation} effect. We show that the existence of regular trajectories poses restrictions to the quantum relaxation process, and we give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.
A usual assumption in the so-called de Broglie–Bohm approach to quantum dynamics is that the quantum trajectories subject to typical 'guiding' wavefunctions turn to be quite irregular, i.e. chaotic (in the dynamical systems' sense). In this paper, we consider mainly cases in which the quantum trajectories are ordered , i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a Hénon–Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie–Bohm formalism, namely the influence of order on the so-called quantum relaxation effect. We show that the existence of regular trajectories poses restrictions on the quantum relaxation process and give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.
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.
In this thesis, properties of particle trajectories associated with the de Broglie-Bohm causal interpretation of quantum mechanics are studied. These trajectories are shown to exhibit deterministic chaos and adiabatic invariance under certain conditions. The very basic elements of the causal interpretation are presented in the first chapter. These include the equations of motion for the particle and the quantum potential. A brief discussion of the philosophically agreeable features of the theory is also included. In Chapter 2, properties of chaotic systems are studied. We define deterministic chaos for a flow and present methods for calculating the maximum Lyapunov exponent. The properties of the different types of systems and the conditions that lead to chaos in these systems are analyzed. We study in detail the specific example of the two-dimensional harmonic oscillator in Chapter 3. We find that different types of trajectories include those which are periodic and chaotic. The necessary conditions for obtaining chaos are determined for a superposition of stationary states. Systems which are qualitatively similar to the harmonic oscillator are covered in Chapter 4. These include the two-dimensional infinite well, an infinite well bisected by a finite barrier, and a Rydberg atom in an external electromagnetic field. In Chapter 5, the effect of a spin 1/2 wavefunction is considered. The causal equations of motion for a spin 1/2 particle are introduced. We find that chaotic trajectories are easily obtained. The causal analogue of the geometric phase is defined in Chapter 6. This phase is shown to be an adiabatic invariant for periodic trajectories. We define the geometric frequency for both periodic and aperiodic trajectories. Finally, in Chapter 7 we examine trajectories associated with stationary states. We define necessary conditions for chaos to arise in the trajectories. The properties of entangled boson and fermion systems are analyzed.
We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule P=|Ψ|2 by initial distributions of Bohmian particles with P0≠|Ψ0|2.
Vortices are known to play a key role in the dynamics of the quantum trajectories defined within the framework of the de Broglie–Bohm formalism of quantum mechanics. It has been rigourously proved that the motion of a vortex in the associated velocity field can induce chaos in these trajectories, and numerical studies have explored the rich variety of behaviors that due to their influence can be observed. In this paper, we go one step further and show how the theory of dynamical systems can be used to construct a general and systematic classification of such dynamical behaviors. This should contribute to establish some firm grounds on which the studies on the intrinsic stochasticity of Bohm's quantum trajectories can be based. An application to the two-dimensional isotropic harmonic oscillator is presented as an illustration.
The de Broglie–Bohm theory is about non-relativistic point-particles that move deterministically along trajectories. The theory reproduces the predictions of standard quantum theory, given that the distribution of particles over an ensemble of systems, all described by the same wavefunction ψ, equals the quantum equilibrium distribution |ψ| 2 . Numerical simulations done by Valentini and Westman (2005 Proc. R. Soc. A 461 253) have illustrated that non-equilibrium particle distributions may relax to quantum equilibrium after some time. Here we consider non-equilibrium distributions and their relaxation properties for a particular class of trajectory theories (first studied in detail by Deotto and Ghirardi (1998 Found. Phys. 28 1)) that are empirically equivalent to the de Broglie–Bohm theory in quantum equilibrium. In the examples we studied of such theories, we found a speed-up of the relaxation, compared to the ordinary de Broglie–Bohm theory. Hence non-equilibrium predictions that depend strongly on relaxation properties, such as those studied recently by Valentini, may vary across different trajectory theories. As such, these theories might be experimentally distinguishable.