  We propose a class of cellular automata for the Hamiltonian of a free
particle. It is based on a two-step unitary evolution operator in discrete time
and space. Various experiments with one and two-dimensional cellular automata
are used to analyze 1) phase velocities of plane waves, 2) dispersion and group
velocities of wavepackets, 3) energy levels of infinite potential wells and
harmonic oscillators, and 4) interference from double-slit diffraction. Some of
the differences between their known (analytical) results and the
cellular-automata approximations are intriguing.
