Quantum fractals in boxes

A quantum wave with probability density P(r, t) = \Psi(r, t)\(2), confined by Dirichlet boundary conditions in a D-dimensional box of arbitrary shape and finite surface area, evolves from the uniform state Psi(r, 0) = 1. For almost all positions r = x(1), x(2)...x(D), the graph of the evolution of P is a fractal curve with dimension D-time = 7/4. For almost all times t, the graph of the spatial probability density P is a fractal hypersurface with dimension D-space = D + 1/2. 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 D - 1 + gamma (0 less than or equal to gamma < 1), simple arguments suggest that the dimension of the time fractal is D-time = (7 + gamma)/4, and that of the space fractal is D-space = D + 1/2 + gamma/2.