  In this work, we study the fluctuation relation and the second law of
thermodynamics within a quantum linear oscillator externally driven over the
period of time t = tau. To go beyond the standard approach (the two-point
projective measurement one) to this subject and also render it discussed in
both quantum and classical domains on the single footing, we recast this
standard approach in terms of the Wigner function and its propagator in the
phase space (x,p). With the help of the canonical transformation from (x,p) to
the angle-action coordinates (\phi,I), we can then derive a measurement-free
(classical-like) form of the Crooks fluctuation relation in the Wigner
representation. This enables us to introduce the work W_{I_0,I_{tau}}
associated with a single run from (I_0) to (I_{tau}) over the period tau, which
is a quantum generalization of the thermodynamic work with its roots in the
classical thermodynamics. This quantum work differs from the energy difference
e_{I_0,I_{tau}} = e(I_{tau}) - e(I_0) unless beta, hbar --> 0. Consequently, we
will obtain the quantum second-law inequality Delta F_{beta} \leq <W>_{P} \leq
<e>_{P} = Delta U, where P, Delta F_{beta}, and <W>_P denote the work
(quasi)-probability distribution, the free energy difference, and the average
work distinguished from the internal energy difference Delta U, respectively,
while <W>_P --> Delta U in the limit of beta, hbar --> 0 only. Therefore, we
can also introduce the quantum heat Q_q = Delta U - W even for a thermally
isolated system, resulting from the quantum fluctuation therein. This is a more
fine-grained result than <W>_P = Delta U obtained from the standard approach.
Owing to the measurement-free nature of the thermodynamic work W_{I_0,I_{tau}},
our result can also apply to the (non-thermal) initial states rho_0 = (1-gamma)
rho_{beta} + gamma sigma with sigma \ne rho_{beta}.
