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arXiv:quant-ph/0412050 · Journal of Physics A Mathematical and General

A Bohmian approach to quantum fractals

Ángel S. Sanz

Published Jun 15, 2005• 15 citations
arXivHTMLPDF

Abstract

A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory-based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well-defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.

DOI
10.1088/0305-4470/38/26/013

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Comment on: Bohmian mechanics with complex action : A new trajectory-based formulation of quantum mechanics

Ángel S. Sanz, Salvador Miret-Artés

In a recent publication, Goldfarb et al. have proposed a new trajectory scheme to solve quantum problems: Bohmian mechanics with complex action BOMCA . According to these authors, their contribution represents “a novel formulation of Bohmian mechanics.” In our opinion, rather than a new formulation of Bohmian mechanics, the proposal of Goldfarb et al. consists of a novel quantum computational scheme, which is based on solving the complex trajectory equations resulting from an already known complex version of Bohmian dynamics. On the other hand, Goldfarb et al. also claimed that their numerical scheme describes “fully local complex quantum trajectories,” with the reward of a significantly higher degree of localization than in the conventional Bohmian mechanics by lowering the magnitude of the quantum potential. At least from a theoretical and interpretative point of view, we consider that it is very important to clarify what “fully local quantum trajectories” means. Bohmian mechanics is a nonlocal theory because standard quantum mechanics is a nonlocal theory. This nonlocality manifests via the presence of a quantum potential, and therefore the corresponding quantum trajectories will display features that reflect such nonlocality. Something similar should also occur for any trajectory-based approach that intends to be equivalent to standard quantum mechanics. In this Comment we would like to provide a general discussion on these two aspects arising from the work of Goldfarb et al., which can be of much interest to the Chemical Physics community. In particular, we would like to stress the special interest relying on how the term locality should be employed. The approach of Goldfarb et al. starts with the timedependent quantum Hamilton-Jacobi equation. This equation was considered by Leacock and Padgett in 1983 within the context of the quantum transformation theory as a way to obtain bound-state energy levels of quantum systems with no need to calculate the corresponding eigenfunctions. According to Leacock and Padgett, the quantum Hamilton-Jacobi equation can be either stated as a postulate as done by these authors or derived from the Schrodinger equation through a simple connection formula. In the second case, the quantum action corresponds to the complex phase of the wave function when the latter is expressed as a pure complex exponential function. This relationship allows one to pass straightforwardly from the quantum Hamilton-Jacobi formalism to the standard one in terms of the time-dependent Schrodinger and vice versa . Although both the Ansatz and the formulation used by Leacock and Padgett are similar to those considered in the standard quantum WKB semiclassical approximation, as pointed out by these authors, their approach conceptually differs from the WKB one. Within Bohmian mechanics, energy eigenstates present an inconvenience: they give rise to zero velocity fields. Hence, when particles are described by this type of states, they will remain steady motionless . To overcome this problem, Floyd and Faraggi and Matone developed timeindependent quantum Hamilton-Jacobi-like formulations starting from real bipolar Ansatze, though they did not claim full equivalence with standard quantum mechanics regarding their predictions. Later, John proposed a timedependent complex quantum trajectory formalism based on the same connection formula mentioned by Leacock and Padgett to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. This “modified de Broglie–Bohm approach to quantum mechanics,” as termed by John, is formally equivalent to the BOMCA formulation of Goldfarb et al. Nevertheless, we would like to specify that, due to the fundamental theoretical nature of John’s contribution, a separate calculation of the wave function is required in order to obtain the corresponding trajectories John makes use of the “guiding wave” idea from Bohmian mechanics . Meanwhile, in the version of Goldfarb et al. the modified de Broglie–Bohm approach is formulated in terms of a closed set of equations in the spirit of quantum hydrodynamical formalisms , which avoids the calculation of the wave functions, thus providing a new, practical use to such an approach. Furthermore, within the approach of Goldfarb et al., the expression for the quantum force happens to be typically much smaller than in standard Bohmian mechanics. On the other hand, we would also like to note that a related formulation has also been used by Chou and Wyatt, though they developed a different, novel numerical scheme to tackle the problem of bound-energy levels and scattering in one dimension. When trying to understand the quantum-classical correspondence from the perspective of a WKB-like formulation of Bohmian mechanics WKB-BM , Sanz and co-workers reached the complex quantum HamiltonJacobi equation the same year as John, but independently. The WKB-BM formulation constitutes an excellent, compreTHE JOURNAL OF CHEMICAL PHYSICS 127, 197101 2007

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A quantum fractal is a wavefunction with a real and an imaginary part continuous everywhere, but differentiable nowhere. This lack of differentiability has been used as an argument to deny the general validity of Bohmian mechanics (and other trajectory-based approaches) in providing a complete interpretation of quantum mechanics. Here, this assertion is overcome by means of a formal extension of Bohmian mechanics based on a limiting approach. Within this novel formulation, the particle dynamics is always satisfactorily described by a well-defined equation of motion. In particular, in the case of guidance under quantum fractals, the corresponding trajectories will also be fractal.
The trajectory-based formalism of Bohmian mechanics constitutes an alternative (but equivalent at a predictive level) approach to the standard formulation of quantum mechanics. Here we show the advantages of this formalism in providing both an accurate description and a novel interpretation when it is applied to different phenomena of interest in elastic particle scattering, such as diffraction by “hard” and “soft” slits, and quantum vortical dynamics due to the presence of single adsorbates. These problems illustrate fairly well how quantum trajectories are able to satisfactorily reproduce the main features of real scattering experiments as well as to provide a causal insight of the underlying dynamics
A solution ψψψ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field ℏℑ(▽ψ/mψ).\hbar \Im \left( \bigtriangledown ψ/mψ\right) .ℏℑ(▽ψ/mψ). In the case of one specific non-differentiable weak solution ΨΨΨ we show how Bohmian trajectories can be obtained for ΨΨΨ from the trajectories of a sequence Ψn→Ψ.Ψ_{n}\rightarrow Ψ.Ψn​→Ψ. (For any real ttt the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors ΨΨΨ in the Hilbert space on which the Schrödinger evolution % Ψ\mapsto e^{-iht}Ψ acts.
Bohmian mechanics is a deterministic theory of quantum mechanics that is based on a set of n velocity functions for n particles, where these functions depend on the wavefunction from the n-body time-dependent Schroedinger equation. It is well know that Bohmian mechanics is not applicable to stationary states, since the velocity field for stationary states is the zero function. Recently, an alternative to Bohmian mechanics has been formulated, based on a conservation of energy equation, where the velocity fields are not the zero function, but this formalism is only applicable to stationary states with real valued wavefunctions. In this paper, Bohmian mechanics is merged with the alternative to Bohmian mechanics. This is accomplished by introducing an interpretation of the Bohm quantum potential. The final formalism gives dynamic particles for all states, including stationary states. The final main working equation contains two kinetic energy terms and a term that contains a factor that can be interpreted as a pressure. The derivation is a simple n-body generalization of the recent generalization, or refinement, of the Madelung equations.
Trajectory-based approaches to quantum mechanics include the de Broglie–Bohm interpretation and Nelson's stochastic interpretation. It is shown that the usual route to establishing the validity of such interpretations, via a decomposition of the Schrödinger equation into a continuity equation and a modified Hamilton–Jacobi equation, fails for some quantum states. A very simple example is provided by a quantum particle in a box, described by a wavefunction that is initially uniform over the interior of the box. For this example, there is no corresponding continuity or modified Hamilton–Jacobi equation, and the space-time dependence of the wavefunction has a known fractal structure. Examples with finite average energies are also constructed.
In a recent publication, Goldfarb et al. have proposed a new trajectory scheme to solve quantum problems: Bohmian mechanics with complex action BOMCA . According to these authors, their contribution represents “a novel formulation of Bohmian mechanics.” In our opinion, rather than a new formulation of Bohmian mechanics, the proposal of Goldfarb et al. consists of a novel quantum computational scheme, which is based on solving the complex trajectory equations resulting from an already known complex version of Bohmian dynamics. On the other hand, Goldfarb et al. also claimed that their numerical scheme describes “fully local complex quantum trajectories,” with the reward of a significantly higher degree of localization than in the conventional Bohmian mechanics by lowering the magnitude of the quantum potential. At least from a theoretical and interpretative point of view, we consider that it is very important to clarify what “fully local quantum trajectories” means. Bohmian mechanics is a nonlocal theory because standard quantum mechanics is a nonlocal theory. This nonlocality manifests via the presence of a quantum potential, and therefore the corresponding quantum trajectories will display features that reflect such nonlocality. Something similar should also occur for any trajectory-based approach that intends to be equivalent to standard quantum mechanics. In this Comment we would like to provide a general discussion on these two aspects arising from the work of Goldfarb et al., which can be of much interest to the Chemical Physics community. In particular, we would like to stress the special interest relying on how the term locality should be employed. The approach of Goldfarb et al. starts with the timedependent quantum Hamilton-Jacobi equation. This equation was considered by Leacock and Padgett in 1983 within the context of the quantum transformation theory as a way to obtain bound-state energy levels of quantum systems with no need to calculate the corresponding eigenfunctions. According to Leacock and Padgett, the quantum Hamilton-Jacobi equation can be either stated as a postulate as done by these authors or derived from the Schrodinger equation through a simple connection formula. In the second case, the quantum action corresponds to the complex phase of the wave function when the latter is expressed as a pure complex exponential function. This relationship allows one to pass straightforwardly from the quantum Hamilton-Jacobi formalism to the standard one in terms of the time-dependent Schrodinger and vice versa . Although both the Ansatz and the formulation used by Leacock and Padgett are similar to those considered in the standard quantum WKB semiclassical approximation, as pointed out by these authors, their approach conceptually differs from the WKB one. Within Bohmian mechanics, energy eigenstates present an inconvenience: they give rise to zero velocity fields. Hence, when particles are described by this type of states, they will remain steady motionless . To overcome this problem, Floyd and Faraggi and Matone developed timeindependent quantum Hamilton-Jacobi-like formulations starting from real bipolar Ansatze, though they did not claim full equivalence with standard quantum mechanics regarding their predictions. Later, John proposed a timedependent complex quantum trajectory formalism based on the same connection formula mentioned by Leacock and Padgett to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. This “modified de Broglie–Bohm approach to quantum mechanics,” as termed by John, is formally equivalent to the BOMCA formulation of Goldfarb et al. Nevertheless, we would like to specify that, due to the fundamental theoretical nature of John’s contribution, a separate calculation of the wave function is required in order to obtain the corresponding trajectories John makes use of the “guiding wave” idea from Bohmian mechanics . Meanwhile, in the version of Goldfarb et al. the modified de Broglie–Bohm approach is formulated in terms of a closed set of equations in the spirit of quantum hydrodynamical formalisms , which avoids the calculation of the wave functions, thus providing a new, practical use to such an approach. Furthermore, within the approach of Goldfarb et al., the expression for the quantum force happens to be typically much smaller than in standard Bohmian mechanics. On the other hand, we would also like to note that a related formulation has also been used by Chou and Wyatt, though they developed a different, novel numerical scheme to tackle the problem of bound-energy levels and scattering in one dimension. When trying to understand the quantum-classical correspondence from the perspective of a WKB-like formulation of Bohmian mechanics WKB-BM , Sanz and co-workers reached the complex quantum HamiltonJacobi equation the same year as John, but independently. The WKB-BM formulation constitutes an excellent, compreTHE JOURNAL OF CHEMICAL PHYSICS 127, 197101 2007
This paper introduces a unified fractal wavefunction model by incorporatinga dynamic fractal dimension Df (t), providing novel explanations for wavefunc?tion geometrization, decoherence, and time irreversibility in quantum mechanics.By reinterpreting the physical essence of the wavefunction from the perspectiveof fractal geometry, we propose a fractal Schr¨odinger equation and demonstratethe extensive applicability of fractal wavefunctions in quantum systems throughexperimental validation and numerical simulations. The primary innovations in?clude the introduction of dynamic fractal dimensions, mathematical modeling offractal wavefunctions, and a geometric interpretation of decoherence and time irre?versibility. With supplementary theoretical foundations, experimental designs, andnumerical analyses, this work enhances the academic rigor and credibility of theproposed model.