# 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} }

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…

## 15 Citations

Quantum Interference, Hidden Symmetries: Theory and Experimental Facts

- PhysicsJournal of Physics: Conference Series
- 2018

The concept of quantum superposition is reconsidered and discussed from the viewpoint of Bohmian mechanics, the hydrodynamic formulation of quantum mechanics, in order to elucidate some physical…

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

- Physics
- 2011

(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…

Quantum carpets of a slightly relativistic particle

- Physics
- 2008

We analyze the structures emerging in the spacetime representation of the probability density woven by a slightly relativistic particle caught in a one-dimensional box. In particular, we evaluate the…

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

- Physics
- 2011

We introduce a pedagogical discussion on Bohmian mechanics and its physical implications in connection with the important role played by the quantum phase in the dynamics of quantum processes. In…

Why isn't every physicist a Bohmian?

- Physics
- 2004

This note collects, classifies and evaluates common criticism against the de Broglie Bohm theory, including Ockham's razor, asymmetry in the de Broglie Bohm theory, the ``surreal trajectory''…

Boundary bound diffraction: a combined spectral and Bohmian analysis

- Physics, MathematicsPhysica Scripta
- 2019

The diffraction-like process displayed by a spatially localized matter wave is here analyzed in a case where the free evolution is frustrated by the presence of hard-wall-type boundaries (beyond the…

A causal look into the quantum Talbot effect.

- Medicine, PhysicsThe Journal of chemical physics
- 2007

The authors provide an insightful picture of this nonlocal phenomenon as well as its classical limit in terms of Bohmian mechanics, also showing the causal reasons and conditions that explain its appearance.

Setting up tunneling conditions by means of Bohmian mechanics

- Mathematics, Physics
- 2011

Usually tunneling is established after imposing some matching conditions on the (time-independent) wave function and its first derivative at the boundaries of a barrier. Here an alternative scheme is…

Bound System Dynamics

- Physics
- 2014

Up to now we have considered quantum systems which are not subject to any external force except for open systems. Such systems are usually associated with translational properties. On the contrary,…

N-Slit Interference: Fractals in Near-Field Region, Bohmian Trajectories

- Physics
- 2009

Scattering cold particles on an $N$-slit grating is shown to reproduce an interference pattern, that manifests itself in the near-field region as the fractal Talbot carpet. In the far-field region…

## References

SHOWING 1-10 OF 36 REFERENCES

Incompleteness of trajectory-based interpretations of quantum mechanics

- Physics
- 2004

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, MedicinePhysical 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

- Mathematics
- 1996

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

- Mathematics, 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

- Physics
- 1966

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, MedicineThe 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, MedicineThe 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.