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Zeitschrift für Naturforschung A

Nonlinear Analysis of Individual Quantum Events in a Model with Bohm Trajectories

Kurt Bräuer

Published Dec 1, 1999• 0 citations
Publisher

Abstract

Abstract The quantum mechanical model under consideration describes a particle beam under the influence of an oscillator. Bohm’s causal interpretation of quantum mechanics is used to calculate trajectories of the particles. Individual quantum events are defined by the intersection of the beam particle trajectories and the particle detector. The time sequence of individual quantum events is interpreted as a time series and is analyzed by linear and nonlinear methods, which involves reconstruction and investigation of the system in an embedding space. The Fourier amplitudes and the fill factors show white noise, however the Karhunen-Loeve components indicate the influence of the oscillator on the beam particles. In this model individual quantum events carry information which can be detected by the Karhunen-Loeve ex-pansion.

DOI
10.1515/zna-1999-1201

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Abstract The quantum mechanical model under consideration describes a particle beam under the influence of an oscillator. Bohm’s causal interpretation of quantum mechanics is used to calculate trajectories of the particles. Individual quantum events are defined by the intersection of the beam particle trajectories and the particle detector. The time sequence of individual quantum events is interpreted as a time series and is analyzed by linear and nonlinear methods, which involves reconstruction and investigation of the system in an embedding space. The Fourier amplitudes and the fill factors show white noise, however the Karhunen-Loeve components indicate the influence of the oscillator on the beam particles. In this model individual quantum events carry information which can be detected by the Karhunen-Loeve ex-pansion.
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.
In the so called ‘causal’ interpretation of quantum mechanics, an electron is considered as a particle and such particle is influenced not only by a classical but also by a so called quantum potential. This idea was developed by Professor Bohm in an important paper. In this paper we use some of the basics of this interpretation in a financial option pricing environment. The causal interpretation allows for trajectories. Path breaking work by Professors Bohm and Hiley and Khrennikov and Choustova have made that the causal interpretation is a step closer to potential applications in social science. In this paper we consider the wave function as a wave of information. We consider the gradient of the phase of this wave function and show how the option price could be influenced by this gradient.
A brief account of the world view of classical physics is given first. We then recapitulate as to why the Copenhagen interpretation of the quantum mechanics had to renounce most of the attractive features of the clasical world view such as a causal description, locality, scientific realism and introduce a fundamental distinction between system and apparatus. The crucial role is played in this by the Bohr's insistence on the wavefunction providing the most complete description possible for an even individual system. The alternative of introducing extra dynamical variables, called hidden variables, in addition to the wavefunction of the system so as to be able to retain at least some of the desirable features of classical physics, is then explored. The first such successful attempt was that of Bohm in 1952 who showed that a realistic interpretation of the quantum mechanics can be given which maintains a causal description as well as does not treat systems and measuring appeartus differently. We begin with the construction of the Bohm's theory. He introduces particle positions as the hidden variables. The particle positions play a special role in Bohm theory. The particle trajectories are guided by the wavefunction. The Bohm theory is deterministic. The probability enters through a special assumption, ``quantum equilibrium'' hypothesis, for the initial conditions on the ensemble of particle trajectories. The ``wave or particle'' dilemma is resolved by a ``wave and particle'' resolution. The measurements in Bohm theory can be described without mysticism. Bohm's theory is however nonlocal.
In relation to this debate perhaps the greatest significance and contribution of Bohm's causal interpretation of quantum mechanics 3 is that it not only exposes the arbitrary philosophical assumptions underlying the claims of the Copenhagen interpretation but also brings into relief the essentially new content of quantum mechanics, which is reflected in different ways in Bohr's interpretation. Indeed the claim that the quantum formalism itself requires us not only to abandon the quest for explanation of quantum phenomena but also the concepts of causality, continuity and the objective reality of individual micro-objects, is shown to be false. However the existence of the single counter-interpretation proposed by Bohm constitutes sufficient grounds for rejecting the absolute and final necessity of complementary description and indeterminacy, along with the inherent unanalysable and closed nature of quantum phenomena.
In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.