2000/05/31 by Daniel Wojcik, Daniel K. Wójcik, Iwo Bialynicki-Birula +3
Mathematics · Physics and Astronomy · #Box counting #Class (philosophy) #Computer science #Dimension (graph theory) #Fractal #Fractal analysis #Fractal dimension #Free particle #Harmonic oscillator #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Particle in a box #Physics #Pure mathematics #Quantum #Quantum chaos and dynamical systems #Quantum harmonic oscillator #Quantum mechanics #Space (punctuation) #Statistical physics #Theoretical and Computational Physics #Wave function #nlin.CD #quant-ph
paper · pdf · doi:10.1103/physrevlett.85.5022
published as Phys. Rev. Lett. 85, 5022 (2000) · 4 pages, 8 figures
arxiv created 2000/09/20 · openalex publication_date 2000/12/11 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Pt(x)\phantom\rule0ex0ex=\phantom\rule0ex0ex|\ensuremathΨ(x,t)|2 is shown not to change during the time evolution. We prove a universal relation Dt\phantom\rule0ex0ex=\phantom\rule0ex0ex1+Dx/2 linking the dimensions of space cross sections Dx and time cross sections Dt of the fractal quantum carpets.