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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 Pₓ (x) 0ex{0ex}=0ex{0ex}| (x, t) |^2 is shown not to change during the time evolution. We prove a universal relation Dₓ{0ex{0ex}=0ex{0ex}1+D}ₗ/2 linking the dimensions of space cross sections Dₗ and time cross sections Dₓ of the fractal quantum carpets.
Wójcik et al. (Mon,) studied this question.