  It is proved here that, as a consequence of the unitary quantum evolution,
the expectation value of a properly defined quantum entropy operator (as
opposed to the non-evolving von Neumann entropy) can only increase during non
adiabatic transformations and remains constant during adiabatic ones. Thus
Clausius formulation of the second law is established as a theorem in quantum
mechanics, in a way that is equivalent to the previously established
formulation in terms of minimal work principle [A. E. Allahverdyan and T. M.
Nieuwenhuizen, Phys. Rev. E 71, 046107 (2005)]. The corresponding Quantum
Mechanical Principle of Entropy Increase is then illustrated with an exactly
solvable example, namely the driven harmonic oscillator. Attention is paid to
both microcanonical and canonical initial condition. The results are compared
to their classical counterparts.

Quantum Physics;Statistical Mechanics