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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 Pₜ(x)0ex0ex=0ex0ex|Ψ(x,t)|² is shown not to change during the time evolution. We prove a universal relation Dₜ0ex0ex=0ex0ex1+Dₓ/2 linking the dimensions of space cross sections Dₓ and time cross sections Dₜ of the fractal quantum carpets.
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Wójcik et al. (2000) studied this question.
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