Time Evolution of Quantum Fractals

Abstract

We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential and free particle. The box-counting dimension of the probability density $P_t(x)=|\Psi (x,t){|}^2$ is shown not to change during the time evolution. We prove a universal relation $D_t=1+D_x/2$ linking the dimensions of space cross-sections $D_x$ and time cross-sections $D_t$ of the fractal quantum carpets.

4 pages, 8 figures

References in corpus (2)

• Some properties of WKB series
• Unravelling quantum carpets: a travelling wave approach

Cited by in corpus (22)

• On the quantification of entanglement in infinite-dimensional quantum systems
• A causal look into the quantum Talbot effect
• Canonical Quantization and Impenetrable Barriers
• Quantum Iterated Function Systems
• Unusual quantum states: nonlocality, entropy, Maxwell's daemon, and fractals
• A transfer matrix method for the analysis of fractal quantum potentials
• Incompleteness of trajectory-based interpretations of quantum mechanics
• Why isn't every physicist a Bohmian?
• Fractal Dynamics in Chaotic Quantum Transport
• A Bohmian approach to quantum fractals
• Analyticity of Quantum States in One-Dimensional Tight-Binding Model
• Chains with Fractal Dispersion Law
• Transition from order to chaos in reduced quantum dynamics
• Fermionic quantum carpets: From canals and ridges to solitonlike structures
• Level-spectra Statistics in Planar Fractal Tight-Binding Models
• Single- to many-body crossover of a quantum carpet
• Limits of fractality: Zeno boxes and relativistic particles
• Classical evolution of fractal measures on the lattice
• A Subtlety of the Schrödinger Picture Dynamics
• Finite Size Universe or Perfect Squash Problem v2
• Bound Eigenstate dynamics under a sudden shift of the well's wall
• Fractality of certain quantum states