  Quantum energy inequalities (QEIs) are lower bounds on the averaged energy
density of a quantum field. They have been proved for various field theories in
general curved spacetimes but the explicit lower bound is not easily calculated
in closed form. In this paper we study QEIs for the massless minimally coupled
scalar field in four-dimensional Minkowski spacetime along stationary
worldlines - curves whose velocity evolves under a 1-parameter Lorentz subgroup
-- and find closed expressions for the QEI bound, in terms of curvature
invariants of the worldline. Our general results are illustrated by specific
computations for the six protoypical stationary worldlines. When the averaging
period is taken to infinity, the QEI bound is consistent with a constant energy
density along the worldline. For inertial and uniformly linearly accelerated
worldlines, this constant value is attained by the Minkowski and Rindler
vacuums respectively. It is an open question as to whether the bounds for other
stationary worldlines are attained by other states of interest.
