Unit function in a Dirichlet box

A Dirichlet box is a region of space with inpenetrable walls. The walls can be described by boundary conditions. Here it is assumed that the initial wave function describes a uniform position distribution inside the box. Hence the initial function (at time t=0) is the function which is everywhere equal to the constant 1. The time evolution is governed by the Dirichlet boundary conditions which immediately force the wave function to be zero at the boundary points. The evolution of the wave packet is periodic in time. This movie only shows a small fraction of the time-period.

In "Visual Quantum Mechanics" it is shown that the wave function is a step function for all times t for which t/T is a rational number (T being the period of the motion).

Here you can see a space-time diagram of the process (a "quantum carpet")

More on the theory of the particle in a box can be found in the following article:

Berry, M V, J.Phys.A 29 6617-6629, ‘Quantum fractals in boxes’. (pdf)