  We have calculated the distribution of work $W$ done on a 1-d harmonic
oscillator that is initially in canonical equilibrium at temperature $T$, then
thermally isolated and driven by an arbitrary time-dependent cyclic spring
constant $\kappa(t)$, and demonstrated that it satisfies $P(W)=\exp(\beta
W)P(-W)$, where $\beta=1/k_{B}T$, in both classical and quantum dynamics. This
differs from the celebrated Crooks relation of nonequilibrium thermodynamics,
since the latter relates distributions for forward and backward protocols of
driving. We show that it is a special case of a symmetry that holds for
non-cyclic work processes on the isolated oscillator, and that consideration of
time reversal invariance shows it to be consistent with the Crooks relation. We
have verified that the symmetry holds in both classical and quantum treatments
of the dynamics, but that inherent uncertainty in the latter case leads to
greater fluctuations in work performed for a given process.
