  We study the time evolution of the Boltzmann entropy of a microstate during
the non-equilibrium free expansion of a one-dimensional quantum ideal gas. This
quantum Boltzmann entropy, $S_B$, essentially counts the "number" of
independent wavefunctions (microstates) giving rise to a specified macrostate.
It generally depends on the choice of macrovariables, such as the type and
amount of coarse-graining, specifying a non-equilibrium macrostate of the
system, but its extensive part agrees with the thermodynamic entropy in thermal
equilibrium macrostates. We examine two choices of macrovariables: the
$U$-macrovariables are local observables in position space, while the
$f$-macrovariables also include structure in momentum space. For the quantum
gas, we use a non-classical choice of the $f$-macrovariables. For both choices,
the corresponding entropies $s_B^f$ and $s_B^U$ grow and eventually saturate.
As in the classical case, the growth rate of $s_B^f$ depends on the momentum
coarse-graining scale. If the gas is initially at equilibrium and is then
released to expand to occupy twice the initial volume, the per-particle
increase in the entropy for the $f$-macrostate, $\Delta s_B^f$, satisfies
$\log{2}\leq\Delta s_B^f\leq 2\log{2}$ for fermions, and $0\leq\Delta
s_B^f\leq\log{2}$ for bosons. For the same initial conditions, the change in
the entropy $\Delta s_B^U$ for the $U$-macrostate is greater than $\Delta
s_B^f$ when the gas is in the quantum regime where the final stationary state
is not at thermal equilibrium.
