  Proper time, ideal clocks, and boosts are well understood classically, but
subtleties arise in quantum physics. We show that quantum clocks set in motion
via momentum boosts do not witness classical time dilation. However, using
velocity boosts we find the ideal behaviour in both cases where the quantum
clock and classical observer are set in motion. Without internal state
dependent forces additional effects arise. As such, we derive observed
frequency shifts in ion trap atomic clocks, indicating a small additional
shift, and also show the emergence of non-ideal behaviour in a theoretical
clock model.
