# A Bohmian approach to quantum fractals

@article{Sanz2005ABA,
title={A Bohmian approach to quantum fractals},
author={{\'A}ngel S. Sanz},
journal={Journal of Physics A},
year={2005},
volume={38},
pages={6037-6049}
}
• Á. S. Sanz
• Published 7 December 2004
• Physics
• Journal of Physics A
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…
16 Citations

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## References

SHOWING 1-10 OF 36 REFERENCES
Incompleteness of trajectory-based interpretations of quantum mechanics
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
Time evolution of quantum fractals
• Physics
Physical review letters
• 2000
A universal relation D(t) = 1+Dx/2 linking the dimensions of space cross sections Dx and time cross sections of the fractal quantum carpets is proved.
Quantum fractals in boxes
A quantum wave with probability density , confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state . For almost all
Fractal noise in quantum ballistic and diffusive lattice systems
• Physics
• 2004
We demonstrate fractal noise in the quantum evolution of wave packets moving either ballistically or diffusively in periodic and quasiperiodic tight-binding lattices, respectively. For the ballistic
Quantum equilibrium and the origin of absolute uncertainty
• Physics
• 1992
The quantum formalism is a “measurement” formalism-a phenomenological formalism describing certain macroscopic regularities. We argue that it can be regarded, and best be understood, as arising from
Derivation of the Schrodinger equation from Newtonian mechanics
We examine the hypothesis that every particle of mass $m$ is subject to a Brownian motion with diffusion coefficient $\frac{\ensuremath{\hbar}}{2m}$ and no friction. The influence of an external
Unravelling quantum carpets: a travelling-wave approach
• Physics
• 1999
Generic channel and ridge structures are known to appear in the time-dependent position probability distribution of a one-dimensional quantum particle confined to a box. These structures are shown to
Quantum trajectories in atom-surface scattering with single adsorbates: the role of quantum vortices.
• Physics
The Journal of chemical physics
• 2004
In this work, a full quantum study of the scattering of He atoms off single CO molecules, adsorbed onto the Pt(111) surface, is presented within the formalism of quantum trajectories provided by
Causal trajectories description of atom diffraction by surfaces
• Physics
• 2000
The method of quantum trajectories proposed by de Broglie and Bohm is applied to the study of atom diffraction by surfaces. As an example, a realistic model for the scattering of He off corrugated Cu
Quantum trajectories in elastic atom-surface scattering: threshold and selective adsorption resonances.
• Physics
The Journal of chemical physics
• 2005
Both threshold and selective adsorption resonances are explained by means of this quantum trapping considering different space and time scales, and a mapping between each region of the (initial) incoming plane wave and the different parts of the diffraction and resonance patterns can be easily established.