  Quantum dynamics controlled by a time-dependent coupling constant are
studied. It is proven that an energy eigenstate expectation value of work done
by the system in a quench process cannot exceed the work in the corresponding
quasi-static process, if and only if the energy eigenvalue is a concave
function of the coupling constant. We propose this concavity of energy
eigenvalues as a new universal criterion for quantum dynamical systems to
satisfy the second law of thermodynamics. We argue simple universal conditions
on quantum systems for the concavity, and show that every energy eigenvalue is
indeed concave in some specific quantum systems. These results agree with the
maximal work principle for adiabatic quench and quasi-static processes as an
expression of the second law of thermodynamics. Our result gives a simple
example of an integrable system satisfying an analogue to the strong eigenstate
thermalization hypothesis (ETH) with respect to the principle of maximum work.
