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Deterministic chaos and the de Broglie-Bohm causal interpretation of quantum mechanics

R.W. Valentine

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

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.

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Journal of Physics Conference Series•Nov 19, 2014•0 citations
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.
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.
The de Broglie-Bohm causal interpretation of quantum mechanics is discussed, and applied to the hydrogen atom in several contexts. Prominent critiques of the causal program are noted and responses are given; it is argued that the de Broglie-Bohm theory is of notable interest to physics. Using the causal theory, electron trajectories are found for the conventional Schrödinger, Pauli and Dirac hydrogen eigenstates. In the Schrödinger case, an additional term is used to account for the spin; this term was not present in the original formulation of the theory but is necessary for the theory to be embedded in a relativistic formulation. In the Schrödinger, Pauli and Dirac cases, the eigenstate trajectories are shown to be circular, with electron motion revolving around the z-axis. Electron trajectories are also found for the 1s-2p0 transition problem under the Schrödinger equation; it is shown that the transition can be characterized by a comparison of the trajectory to the relevant eigenstate trajectories. The structures of the computed trajectories are relevant to the question of the possible evolution of a quantum distribution towards the standard quantum distribution (quantum equilibrium); this process is known as quantum relaxation. The transition problem is generalized to include all possible transitions in hydrogen stimulated by semi-classical radiation, and all of the trajectories found are examined in light of their implications for the evolution of the distribution to the standard distribution. Several promising avenues for future research are discussed.
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.
Quantum trajectories defined in the de Broglie-Bohm theory provide a causal way to interpret physical phenomena. In this letter, we use this formalism to analyze the short-time dynamics induced by unstable periodic orbits in a classically chaotic system, a situation in which scars are known to play a very important role. We find that the topologies of the quantum orbits are much more complicated than that of the scarring and associated periodic orbits, since the former have quantum interference built in. Thus scar wave functions are necessary to analyze the corresponding dynamics. Moreover, these topologies imply different return routes to the vicinity of the initial positions, and this reflects in the existence of different contributions in each peak of the survival probability function.
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.
Bohmian mechanics is a causal interpretation of quantum mechanics in which there are well-defined particle trajectories guided by the wave function. Within this framework, it is shown that certain classical integrable systems can exhibit both periodic and chaotic behaviours. In this paper, we extend the work of Borondo et al. by investigating the behaviours of commensurate two-dimensional harmonic oscillator systems with different ratios of frequencies. We show that the system may produce chaotic behavior of Bohmian trajectories to be dependenton frequencies ratio. This result is illustrated numerically in computer experiments displaying the Bohmian trajectories.