  We propose that a quantum particle in a potential in one space dimension can
be described by a probabilistic cellular automaton. While the simple updating
rule of the automaton is deterministic, the probabilistic description is
introduced by a probability distribution over initial conditions. The proposed
automaton involves right- and left-movers, jumping from one cell to a
neighboring one. They change their direction of motion at each randomly
distributed disorder or scattering point. The continuum limit of an infinite
number of cells yields a Dirac equation, and in the non-relativistic limit the
familiar Schr\"odinger equation, with potential determined by the
spacetime-distribution of scattering points. These equations describe the time
evolution of the probabilistic information for the position of the particle.
All quantum rules for observables, both for discrete possible measurement
values and continuous expectation values, follow from the classical statistical
laws.
