The second law of Thermodynamics as a theorem in quantum mechanics

We treat a quantum mechanical system with certain general properties which are expected to be common in macroscopic quantum systems. Starting from a PURE initial state (which may not describe an equilibrium) in which energy is mildly concentrated at a single value, we consider a time evolution determined by a time-dependent Hamiltonian as a model of an adiabatic operation in thermodynamics. We take a family of operations with the same procedure and various ``waiting times.‘’ Then the Minimum Work Principle is rigorously proved for almost all choices of the waiting time.