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Analyzing the quantum phase with quantum trajectories: A step towards the creation of a Bohmian thinking

Ángel S. Sanz

Published Apr 7, 2011• 1 citation
OpenAlex

Abstract

(Dated: April 8, 2011)Standard quantum mechanics relies on a series of postulates which constitute its fundamentalpillars. In Bohmian mechanics such postulates appear as a natural consequence of the statisticaltreatment of a trajectory-based theory (just as in classical mechanics). This is a reflection onBohmian mechanics and its physical implication at both a pedagogical and applied level whenanalyzing the role of quantum phase in different contexts of quantum mechanics. In particular, tocarry out this study, here we have mainly focused on the phenomena of quantum coherence andquantum interference, which are central to many different fundamental and applied problems ofinterest at present.I. INTRODUCTION

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(Dated: April 8, 2011)Standard quantum mechanics relies on a series of postulates which constitute its fundamentalpillars. In Bohmian mechanics such postulates appear as a natural consequence of the statisticaltreatment of a trajectory-based theory (just as in classical mechanics). This is a reflection onBohmian mechanics and its physical implication at both a pedagogical and applied level whenanalyzing the role of quantum phase in different contexts of quantum mechanics. In particular, tocarry out this study, here we have mainly focused on the phenomena of quantum coherence andquantum interference, which are central to many different fundamental and applied problems ofinterest at present.I. INTRODUCTION
The origin of quantum interference characteristic of bound nonlinear systems is investigated within the Bohmian formulation of time-dependent quantum mechanics. By contrast to time-dependent semiclassical theory, whereby interference is a consequence of phase mismatch between distinct classical trajectories, the Bohmian, fully quantum mechanical expression for expectation values has a quasiclassical appearance that does not involve phase factors or cross terms. Numerical calculations reveal that quantum interference in the Bohmian formulation manifests itself directly as sharp spatial/temporal variations of the density surrounding kinky trajectories. These effects are most dramatic in regions where the underlying classical motion exhibits focal points or caustics, and crossing of the Bohmian trajectories is prevented through extremely strong and rapidly varying quantum mechanical forces. These features of Bohmian dynamics, which constitute the hallmark of quantum interference and are ubiquitous in bound nonlinear systems, represent a major source of instability, making the integration of the Bohmian equations extremely demanding in such situations.
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.
Mermin's "shut up and calculate!" somehow summarizes the most widely accepted view on quantum mechanics. This conception has led to a rather constraining way to think and understand the quantum world. Nonetheless, a closer look at the principles and formal body of this theory shows that, beyond longstanding prejudices, there is still room enough for alternative tools. This is the case, for example, of Bohmian mechanics. As it is discussed here, there is nothing contradictory or wrong with this hydrodynamical representation, which enhances the dynamical role of the quantum phase to the detriment (to some extent) of the probability density. The possibility to describe the evolution of quantum systems in terms of trajectories or streamlines is just a direct consequence of the fact that Bohmian mechanics (quantum hydrodynamics) is just a way to recast quantum mechanics in the more general language of the theory of characteristics. Misconceptions concerning Bohmian mechanics typically come from the fact that many times it is taken out of context and considered as an alternative theory to quantum mechanics, which is not the case. On the contrary, an appropriate contextualization shows that Bohmian mechanics constitutes a serious and useful representation of quantum mechanics, at the same level as any other quantum picture, such as Schrödinger's, Heisenberg's, Dirac's, or Feynman's, for instance. To illustrate its versatility, two phenomena will be briefly considered, namely dissipation and light interference.
Since its inception Bohmian mechanics has been generally regarded as a hidden-variable theory aimed at providing an objective description of quantum phenomena. To date, this rather narrow conception of Bohm’s proposal has caused it more rejection than acceptance. Now, after 65 years of Bohmian mechanics, should still be such an interpretational aspect the prevailing appraisal? Why not favoring a more pragmatic view, as a legitimate picture of quantum mechanics, on equal footing in all respects with any other more conventional quantum picture? These questions are used here to introduce a discussion on an alternative way to deal with Bohmian mechanics at present, enhancing its aspect as an efficient and useful picture or formulation to tackle, explore, describe and explain quantum phenomena where phase and correlation ( entanglement ) are key elements. This discussion is presented through two complementary blocks. The first block is aimed at briefly revisiting the historical context that gave rise to the appearance of Bohmian mechanics, and how this approach or analogous ones have been used in different physical contexts. This discussion is used to emphasize a more pragmatic view to the detriment of the more conventional hidden-variable (ontological) approach that has been a leitmotif within the quantum foundations. The second block focuses on some particular formal aspects of Bohmian mechanics supporting the view presented here, with special emphasis on the physical meaning of the local phase field and the associated velocity field encoded within the wave function. As an illustration, a simple model of Young’s two-slit experiment is considered. The simplicity of this model allows to understand in an easy manner how the information conveyed by the Bohmian formulation relates to other more conventional concepts in quantum mechanics. This sort of pedagogical application is also aimed at showing the potential interest to introduce Bohmian mechanics in undergraduate quantum mechanics courses as a working tool rather than merely an alternative interpretation.
Quantum mechanics is often praised as a theory which unifies classical mechanics and classical wave theory. Quanta are said to behave either as particles or waves, depending on the type of experiment they are subjected to. But where in the standard formalism can a pointlike particle structure actually be found? Perhaps only to some degree in the reduction postulate applied to position measurements. In reaction to this unsatisfactory state of affairs, Bohmian mechanics introduces a mathematically precise particle concept into quantum mechanical theory. The fuzzy wave functions are supplemented by sharp particle world-lines. Through this additional structure some quantum phenomena like the double slit experiment have lost their mystery.