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 $P_t\left(x\right)=|\Psi (x,t){|}^2$ 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.

• Received 18 May 2000

DOI:https://doi.org/10.1103/PhysRevLett.85.5022