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.
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Berkel et al. (2024) studied this question.
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