  We consider a quantum gas of non-interacting particles confined in the
expanding cavity, and investigate the nature of the non-adiabatic force which
is generated from the gas and acts on the cavity wall. Firstly, with use of the
time-dependent canonical transformation which transforms the expanding cavity
to the non-expanding one, we can define the force operator. Secondly, applying
the perturbative theory which works when the cavity wall begins to move at time
origin, we find that the non-adiabatic force is quadratic in the wall velocity
and thereby does not break the time-reversal symmetry, in contrast with the
general belief. Finally, using an assembly of the transitionless quantum
states, we obtain the nonadiabatic force exactly. The exact result justifies
the validity of both the definition of force operator and the issue of the
perturbative theory. The mysterious mechanism of nonadiabatic transition with
use of transitionless quantum states is also explained. The study is done on
both cases of the hard-wall and soft-wall confinement with the time-dependent
confining length.
