Abstract
A quantum wave with probability density 
, confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state 
. For almost all positions 
, the graph of the evolution of P is a fractal curve with dimension 
. For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension 
. When D = 1, there are, in addition to these generic time and space fractals, infinitely many special `quantum revival' times when P is piecewise constant, and infinitely many special spacetime slices for which the dimension of P is 5/4. If the surface of the box is a fractal with dimension 
, simple arguments suggest that the dimension of the time fractal is 
, and that of the space fractal is 
.