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 Pt(x)=|(x,t)|2 is shown not to change during the time evolution. We prove a universal relation Dt=1+Dx/2 linking the dimensions of space cross-sections Dx and time cross-sections Dt of the fractal quantum carpets.
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