  The Dirac equation is a cornerstone in the history of physics, merging
successfully quantum mechanics with special relativity, providing a natural
description of the electron spin and predicting the existence of anti-matter.
Furthermore, it is able to reproduce accurately the spectrum of the hydrogen
atom and its realm, relativistic quantum mechanics, is considered as the
natural transition to quantum field theory. However, the Dirac equation also
predicts some peculiar effects such as Klein's paradox and Zitterbewegung, an
unexpected quivering motion of a free relativistic quantum particle first
examined by Schr\"odinger. These and other predictions would be difficult to
observe in real particles, while constituting key fundamental examples to
understand relativistic quantum effects. Recent years have seen an increased
interest in simulations of relativistic quantum effects in different physical
setups, where parameter tunability allows accessibility to different physical
regimes. Here, we perform a proof-of-principle quantum simulation of the
one-dimensional Dirac equation using a single trapped ion, which is set to
behave as a free relativistic quantum particle. We measure as a function of
time the particle position and study Zitterbewegung for different initial
superpositions of positive and negative energy spinor states, as well as the
cross-over from relativistic to nonrelativistic dynamics. The high level of
control of trapped-ion experimental parameters makes it possible to simulate
elegant textbook examples of relativistic quantum physics.
