  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 $P_t(x)=|\Psi(x,t)|^2$ 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.
